var bibbase_data = {"data":"<img src=\"//bibbase.org/img/ajax-loader.gif\" id=\"spinner\"\n  style=\"display: none;\" alt=\"Loading..\" />\n\n<div id=\"bibbase\">\n\n  <script type=\"text/javascript\">\n    var bibbase = {\n      params: {\"bib\":\"https://www.sas.upenn.edu/~htowsner/papers.bib\",\"simplegroups\":\"true\",\"group0\":\"year\",\"css\":\"https://www.sas.upenn.edu/~htowsner/mybibstyle.css\",\"jsonp\":\"1\",\"host\":\"bibbase.org\"},\n      data: [{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Metric fixed point theory and partial impredicativity\",\"volume\":\"381\",\"issn\":\"1471-2962\",\"url\":\"http://dx.doi.org/10.1098/rsta.2022.0012\",\"doi\":\"10.1098/rsta.2022.0012\",\"urlarxiv\":\"https://arxiv.org/abs/2302.08874\",\"number\":\"2248\",\"journal\":\"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"publisher\":\"The Royal Society\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Fernández-Duque\"],\"firstnames\":[\"D.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Shafer\"],\"firstnames\":[\"P.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yokoyama\"],\"firstnames\":[\"K.\"],\"suffixes\":[]}],\"year\":\"2023\",\"month\":\"April\",\"abstract\":\"We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in 𝖱𝖢𝖠0. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between 𝖠𝖳𝖱0 and Π11-𝖢𝖠0. We also exhibit several weakenings of Caristi's theorem that are equivalent to 𝖶𝖪𝖫0 and to 𝖠𝖢𝖠0. \",\"bibtex\":\"@article{FernndezDuque2023,\\n  title = {Metric fixed point theory and partial impredicativity},\\n  volume = {381},\\n  ISSN = {1471-2962},\\n  url = {http://dx.doi.org/10.1098/rsta.2022.0012},\\n  DOI = {10.1098/rsta.2022.0012},\\n  urlarxiv = {https://arxiv.org/abs/2302.08874},\\n  number = {2248},\\n  journal = {Philosophical Transactions of the Royal Society A: Mathematical,  Physical and Engineering Sciences},\\n  publisher = {The Royal Society},\\n  author = {Fernández-Duque,  D. and Shafer,  P. and Towsner,  H. and Yokoyama,  K.},\\n  year = {2023},\\n  month = apr ,\\n  abstract = {    We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in 𝖱𝖢𝖠0. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between 𝖠𝖳𝖱0 and Π11-𝖢𝖠0. We also exhibit several weakenings of Caristi's theorem that are equivalent to 𝖶𝖪𝖫0 and to 𝖠𝖢𝖠0. }\\n}\\n\\n\",\"author_short\":[\"Fernández-Duque, D.\",\"Shafer, P.\",\"Towsner, H.\",\"Yokoyama, K.\"],\"key\":\"FernndezDuque2023\",\"id\":\"FernndezDuque2023\",\"bibbaseid\":\"fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023\",\"role\":\"author\",\"urls\":{\"Paper\":\"http://dx.doi.org/10.1098/rsta.2022.0012\",\"Arxiv\":\"https://arxiv.org/abs/2302.08874\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":13,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"From Saturated Embedding Tests to Explicit Algorithms\",\"year\":\"2023\",\"eprint\":\"arXiv:2306.12239\",\"urlarxiv\":\"https://arxiv.org/abs/2306.12239\",\"abstract\":\"Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form – usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm. We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and Günaydin. \",\"bibtex\":\"@misc{2306.12239,\\nAuthor = {Henry Towsner},\\nTitle = {From Saturated Embedding Tests to Explicit Algorithms},\\nYear = {2023},\\nEprint = {arXiv:2306.12239},\\nurlarxiv = {https://arxiv.org/abs/2306.12239},\\nabstract = {     Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm.\\n    We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and Günaydin. },\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"2306.12239\",\"id\":\"2306.12239\",\"bibbaseid\":\"towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/2306.12239\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":12,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"An Aldous–Hoover Theorem for Radon Distributions\",\"year\":\"2023\",\"eprint\":\"arXiv:2306.03057\",\"urlarxiv\":\"https://arxiv.org/abs/2306.03057\",\"abstract\":\"We show that the Aldous–Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam i̧tealam2023generalizing who showed a similar generalization of the di Finetti–Hewitt-Savage Theorem. \",\"bibtex\":\"@misc{2306.03057,\\nAuthor = {Henry Towsner},\\nTitle = {An Aldous--Hoover Theorem for Radon Distributions},\\nYear = {2023},\\nEprint = {arXiv:2306.03057},\\nurlarxiv = {https://arxiv.org/abs/2306.03057},\\nabstract = {     We show that the Aldous--Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam \\\\cite{alam2023generalizing} who showed a similar generalization of the di Finetti--Hewitt-Savage Theorem. },\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"2306.03057\",\"id\":\"2306.03057\",\"bibbaseid\":\"towsner-analdoushoovertheoremforradondistributions-2023\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/2306.03057\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"doi\":\"10.1142/s0219061322500234\",\"url\":\"https://doi.org/10.1142/s0219061322500234\",\"year\":\"2022\",\"month\":\"December\",\"publisher\":\"World Scientific Pub Co Pte Ltd\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]},{\"firstnames\":[\"Rose\"],\"propositions\":[],\"lastnames\":[\"Weisshaar\"],\"suffixes\":[]},{\"firstnames\":[\"Linda\"],\"propositions\":[],\"lastnames\":[\"Westrick\"],\"suffixes\":[]}],\"title\":\"Borel combinatorics fail in HYP\",\"journal\":\"Journal of Mathematical Logic\",\"urlarxiv\":\"https://arxiv.org/abs/2106.13330\",\"abstract\":\"We characterize the completely determined Borel subsets of HYP as exactly the omega_1^ck subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author. \",\"bibtex\":\"@article{CombinatoricsFails,\\n  doi = {10.1142/s0219061322500234},\\n  url = {https://doi.org/10.1142/s0219061322500234},\\n  year = {2022},\\n  month = dec,\\n  publisher = {World Scientific Pub Co Pte Ltd},\\n  author = {Henry Towsner and Rose Weisshaar and Linda Westrick},\\n  title = {Borel combinatorics fail in HYP},\\n  journal = {Journal of Mathematical Logic},\\n  urlarxiv = {https://arxiv.org/abs/2106.13330},\\n  abstract={     We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author. },\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\",\"Weisshaar, R.\",\"Westrick, L.\"],\"key\":\"CombinatoricsFails\",\"id\":\"CombinatoricsFails\",\"bibbaseid\":\"towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://doi.org/10.1142/s0219061322500234\",\"Arxiv\":\"https://arxiv.org/abs/2106.13330\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"A Removal Lemma for Ordered Hypergraphs\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"note\":\"draft\",\"year\":\"2021\",\"eprint\":\"2101.09769\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.CO\",\"urlarxiv\":\"https://arxiv.org/abs/2101.09769\",\"abstract\":\"We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon–Ben-Eliezer–Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,<) has few induced copies of a small ordered hypergraph (W,H,≺) then there is a small modification G′ so that (V,G′,<) has no induced copies of (W,H,≺). (Note that we do \\\\emphnot need to modify the ordering <.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of σ-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon. \",\"bibtex\":\"@misc{towsner2021removal,\\n      title={A Removal Lemma for Ordered Hypergraphs}, \\n      author={Henry Towsner},\\nnote={draft},\\n      year={2021},\\n      eprint={2101.09769},\\n      archivePrefix={arXiv},\\n      primaryClass={math.CO},\\n      urlarxiv={https://arxiv.org/abs/2101.09769},\\n      abstract={    We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,<) has few induced copies of a small ordered hypergraph (W,H,≺) then there is a small modification G′ so that (V,G′,<) has no induced copies of (W,H,≺). (Note that we do \\\\emph{not} need to modify the ordering <.)\\n      \\n    We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of σ-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon. },\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner2021removal\",\"id\":\"towsner2021removal\",\"bibbaseid\":\"towsner-aremovallemmafororderedhypergraphs-2021\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/2101.09769\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":60,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"title\":\"Hypergraph regularity and higher arity VC-dimension\",\"author\":[{\"firstnames\":[\"Artem\"],\"propositions\":[],\"lastnames\":[\"Chernikov\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"note\":\"draft\",\"year\":\"2020\",\"eprint\":\"2010.00726\",\"archiveprefix\":\"arXiv\",\"primaryclass\":\"math.CO\",\"urlarxiv\":\"https://arxiv.org/abs/2010.00726\",\"abstract\":\"We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels—one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples—and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1. We also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices. \",\"bibtex\":\"@misc{chernikov2020hypergraph,\\n      title={Hypergraph regularity and higher arity VC-dimension}, \\n      author={Artem Chernikov and Henry Towsner},\\nnote={draft},\\n      year={2020},\\n      eprint={2010.00726},\\n      archivePrefix={arXiv},\\n      primaryClass={math.CO},\\n      urlarxiv={https://arxiv.org/abs/2010.00726},\\n      abstract={ We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets.\\nIn the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels---one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples---and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1.\\nWe also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices. },\\n}\\n\\n\",\"author_short\":[\"Chernikov, A.\",\"Towsner, H.\"],\"key\":\"chernikov2020hypergraph\",\"id\":\"chernikov2020hypergraph\",\"bibbaseid\":\"chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/2010.00726\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":21,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Simmons\"],\"firstnames\":[\"W.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Explicit polynomial bounds on prime ideals in polynomial rings over fields\",\"month\":\"August\",\"year\":\"2020\",\"urlarxiv\":\"https://arxiv.org/abs/1808.04805\",\"journal\":\"Pacific Journal of Mathematics\",\"vol\":\"306\",\"issue\":\"2\",\"doi\":\"10.2140/pjm.2020.306.721\",\"pages\":\"721–754\",\"abstract\":\"Suppose I is an ideal of a polynomial ring over a field, I⊆k[x1,…,xn], and whenever fg∈I with degree ≤b, then either f∈I or g∈I. When b is sufficiently large, it follows that I is prime. Schmidt-Göttsch proved that \\\"sufficiently large\\\" can be taken to be a polynomial in the degree of generators of I (with the degree of this polynomial depending on n). However Schmidt-Göttsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on b, polynomial in the degree of the generators of I. We also give a similar bound for detecting maximal ideals in k[x1,…,xn].\",\"bibtex\":\"@article{2018arXiv180804805S,\\n   author = {{Simmons}, W. and {Towsner}, H.},\\n    title = \\\"{Explicit polynomial bounds on prime ideals in polynomial rings over fields}\\\",\\n    month = {August},\\n    year = {2020},\\n    urlarxiv={https://arxiv.org/abs/1808.04805},\\n    journal={Pacific Journal of Mathematics},\\n    vol = {306},\\n    issue = {2},\\n    doi = {10.2140/pjm.2020.306.721},\\n    pages = {721--754},\\n    abstract={Suppose I is an ideal of a polynomial ring over a field, I⊆k[x1,…,xn], and whenever fg∈I with degree ≤b, then either f∈I or g∈I. When b is sufficiently large, it follows that I is prime. Schmidt-G\\\\\\\"ottsch proved that \\\"sufficiently large\\\" can be taken to be a polynomial in the degree of generators of I (with the degree of this polynomial depending on n). However Schmidt-G\\\\\\\"ottsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on b, polynomial in the degree of the generators of I. We also give a similar bound for detecting maximal ideals in k[x1,…,xn].}\\n}\\n\\n\",\"author_short\":[\"Simmons, W.\",\"Towsner, H.\"],\"key\":\"2018arXiv180804805S\",\"id\":\"2018arXiv180804805S\",\"bibbaseid\":\"simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1808.04805\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":16,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yokoyama\"],\"firstnames\":[\"K.\"],\"suffixes\":[]}],\"title\":\"Erdos-Moser and ISigma_2\",\"note\":\"draft\",\"year\":\"2018\",\"month\":\"July\",\"urlarxiv\":\"https://arxiv.org/abs/1807.04452\",\"abstract\":\"The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erdős-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.\",\"bibtex\":\"@misc{1807.04452,\\nauthor = {{Towsner}, H. and {Yokoyama}, K.},\\ntitle=\\\"{Erdos-Moser and ISigma_2}\\\",\\nnote={draft},\\nyear=2018,\\nmonth=jul,\\nurlarxiv={https://arxiv.org/abs/1807.04452},\\nabstract={The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erd\\\\H{o}s-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. \\nSpecifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.}}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\",\"Yokoyama, K.\"],\"key\":\"1807.04452\",\"id\":\"1807.04452\",\"bibbaseid\":\"towsner-yokoyama-erdosmoserandisigma2-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1807.04452\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":14,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Easwaren\"],\"firstnames\":[\"K.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Realism in Mathematics: The Case of the Hyperreals\",\"note\":\"submitted\",\"year\":\"2018\",\"month\":\"June\",\"urlarxiv\":\"https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1\",\"abstract\":\"A traditional question in the philosophy of mathematics is whether abstract mathematical objects really exist, just as planets and atoms and giraffes do, independently of the human mind. Views on this question are often grouped into “platonist”, “nominalist”, and “constructivist” groups depending on whether they state that mathematical objects have their own independent existence, or don’t truly exist at all, or exist but only as constructions of the human mind. This paper will not directly address this question. We expect that to the extent that each of these views is defensible, they will not settle the questions that we are interested in (though there may be some affinities between some possible answers to these questions). Rather, we will focus on two other questions of mathematical realism: the question of which mathematical claims can be taken as meaningful and true within mathematics, and which mathematical ideas can be taken as applying to the physical world as part of a scientific theory. We will focus on these questions for the case of the “hyperreals” of nonstandard analysis, introduced by Abraham Robinson in the 1960’s (which we will describe in greater detail later). Because of their mathematical properties, the hyperreals have attracted attacks and defenses in ways that are uncommon for mathematical entities. Our central claim is that many of the views on both sides of these debates conflate two distinct philosophical questions. Defenders of the hyperreals give realist answers to both questions, while attackers give antirealist answers to both. We defend a realist answer to one question and an antirealist answer to the other. We think the distinction between these two questions is important not just for thinking about the hyperreals, but for other related mathematical entities, and particularly those whose existence depend on the Axiom of Choice.\",\"bibtex\":\"@misc{EaswarenTowsner,\\nauthor = {{Easwaren}, K. and {Towsner}, H.},\\ntitle = \\\"{Realism in Mathematics: The Case of the Hyperreals}\\\",\\nnote={submitted},\\nyear=2018,\\nmonth= jun,\\nurlarxiv={https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1},\\nabstract={A traditional question in the philosophy of mathematics is whether abstract\\nmathematical objects really exist, just as planets and atoms and giraffes do,\\nindependently of the human mind. Views on this question are often grouped into\\n“platonist”, “nominalist”, and “constructivist” groups depending on whether\\nthey state that mathematical objects have their own independent existence, or\\ndon’t truly exist at all, or exist but only as constructions of the human mind.\\nThis paper will not directly address this question. We expect that to the extent\\nthat each of these views is defensible, they will not settle the questions that we\\nare interested in (though there may be some affinities between some possible\\nanswers to these questions). Rather, we will focus on two other questions of\\nmathematical realism: the question of which mathematical claims can be taken\\nas meaningful and true within mathematics, and which mathematical ideas can\\nbe taken as applying to the physical world as part of a scientific theory.\\nWe will focus on these questions for the case of the “hyperreals” of nonstandard\\nanalysis, introduced by Abraham Robinson in the 1960’s (which we\\nwill describe in greater detail later). Because of their mathematical properties,\\nthe hyperreals have attracted attacks and defenses in ways that are uncommon\\nfor mathematical entities. Our central claim is that many of the views on both\\nsides of these debates conflate two distinct philosophical questions. Defenders\\nof the hyperreals give realist answers to both questions, while attackers give\\nantirealist answers to both. We defend a realist answer to one question and\\nan antirealist answer to the other. We think the distinction between these two\\nquestions is important not just for thinking about the hyperreals, but for other\\nrelated mathematical entities, and particularly those whose existence depend\\non the Axiom of Choice.}\\n}\\n\\n\\n\",\"author_short\":[\"Easwaren, K.\",\"Towsner, H.\"],\"key\":\"EaswarenTowsner\",\"id\":\"EaswarenTowsner\",\"bibbaseid\":\"easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":26,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"What do ultraproducts remember about the original structures?\",\"note\":\"draft\",\"year\":\"2018\",\"month\":\"April\",\"urlarxiv\":\"https://arxiv.org/abs/1804.10809\",\"abstract\":\"We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.\",\"bibtex\":\"@misc{1804.10809,\\n   author = {{Towsner}, H.},\\n    title = \\\"{What do ultraproducts remember about the original structures?}\\\",\\n    note={draft},\\n     year = 2018,\\n    month = apr,\\nurlarxiv={https://arxiv.org/abs/1804.10809},\\nabstract={We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"1804.10809\",\"id\":\"1804.10809\",\"bibbaseid\":\"towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1804.10809\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":36,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Bernshteyn\"],\"firstnames\":[\"A.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Delcourt\"],\"firstnames\":[\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Tserunyan\"],\"firstnames\":[\"A.\"],\"suffixes\":[]}],\"title\":\"A short nonalgorithmic proof of the containers theorem for hypergraphs\",\"vol\":\"147\",\"pages\":\"1739–1749\",\"year\":\"2019\",\"urlarxiv\":\"https://arxiv.org/abs/1801.07186\",\"urljournal\":\"https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/\",\"journal\":\"Proc. Amer. Math. Soc\",\"doi\":\"10.1090/proc/14368\",\"abstract\":\"Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogues of a variety of classical problems from combinatorics and number theory. In this paper, we give the first known proof of this theorem that does not involve an inductive construction (i.e., the so-called scythe algorithm). This proof is less than 4 pages long while being entirely self-contained. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. \",\"bibtex\":\"@article{1801.07186,\\n   author = {{Bernshteyn}, A. and {Delcourt}, M. and {Towsner}, H. and {Tserunyan}, A.\\n\\t},\\n    title = \\\"{A short nonalgorithmic proof of the containers theorem for hypergraphs}\\\",\\nvol=147,\\npages={1739--1749},\\n     year = 2019,\\n   urlarxiv = {https://arxiv.org/abs/1801.07186},\\n   urljournal={https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/},\\n   journal = {Proc. Amer. Math. Soc},\\n   doi = {10.1090/proc/14368},\\nabstract={Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogues of a variety of classical problems from combinatorics and number theory. In this paper, we give the first known proof of this theorem that does not involve an inductive construction (i.e., the so-called scythe algorithm). This proof is less than 4 pages long while being entirely self-contained. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets.\\n}\\n}\\n\\n\\n\",\"author_short\":[\"Bernshteyn, A.\",\"Delcourt, M.\",\"Towsner, H.\",\"Tserunyan, A.\"],\"key\":\"1801.07186\",\"id\":\"1801.07186\",\"bibbaseid\":\"bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1801.07186\",\"Journal\":\"https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":22,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"More or Less Uniform Convergence\",\"year\":\"2017\",\"month\":\"September\",\"note\":\"submitted\",\"urlarxiv\":\"https://arxiv.org/abs/1709.05687\",\"abstract\":\"Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals. We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract alpha-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when \\\"proof mining\\\" convergence proofs to obtain quantitative bounds. We show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.\",\"bibtex\":\"@misc{1709.05687,\\nauthor ={Henry Towsner},\\ntitle = {More or Less Uniform Convergence},\\nyear={2017},\\nmonth= sep,\\nnote={submitted},\\nurlarxiv={https://arxiv.org/abs/1709.05687},\\nabstract={Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals.\\n\\nWe give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract alpha-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. \\n\\nThe proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when \\\"proof mining\\\" convergence proofs to obtain quantitative bounds. \\n\\nWe show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"1709.05687\",\"id\":\"1709.05687\",\"bibbaseid\":\"towsner-moreorlessuniformconvergence-2017\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1709.05687\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":7,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Nonstandard Convergence Gives Bounds on Jumps\",\"year\":\"2019\",\"month\":\"August\",\"journal\":\"New York Journal of Mathematics\",\"vol\":\"25\",\"pages\":\"651–667\",\"url\":\"http://nyjm.albany.edu/j/2019/25-31.html\",\"urlarxiv\":\"https://arxiv.org/abs/1705.10355\",\"abstract\":\"If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform \\\"meta-stable\\\" rate of convergence. We show that obtaining a stronger bound—a uniform bound on the number of jumps the sequence makes—is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.\",\"bibtex\":\"@article{1705.10355,\\nauthor = {Henry Towsner},\\ntitle = {Nonstandard Convergence Gives Bounds on Jumps},\\nyear = {2019},\\nmonth = aug,\\njournal = {New York Journal of Mathematics},\\nvol = {25},\\npages = {651--667},\\nurl = {http://nyjm.albany.edu/j/2019/25-31.html},\\nurlarxiv={https://arxiv.org/abs/1705.10355},\\nabstract={If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform \\\"meta-stable\\\" rate of convergence. We show that obtaining a stronger bound---a uniform bound on the number of jumps the sequence makes---is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.}\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"1705.10355\",\"id\":\"1705.10355\",\"bibbaseid\":\"towsner-nonstandardconvergencegivesboundsonjumps-2019\",\"role\":\"author\",\"urls\":{\"Paper\":\"http://nyjm.albany.edu/j/2019/25-31.html\",\"Arxiv\":\"https://arxiv.org/abs/1705.10355\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":9,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"I.\"],\"propositions\":[],\"lastnames\":[\"Goldbring\"],\"suffixes\":[]},{\"firstnames\":[\"B.\"],\"propositions\":[],\"lastnames\":[\"Hart\"],\"suffixes\":[]},{\"firstnames\":[\"H.\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Explicit sentences distinguishing McDuff's II$_1$ factors\",\"urlarxiv\":\"https://arxiv.org/abs/1701.07928\",\"journal\":\"Israel Journal of Mathematics\",\"doi\":\"10.1007/s11856-018-1735-8\",\"year\":\"2018\",\"month\":\"Aug\",\"day\":\"01\",\"volume\":\"227\",\"number\":\"1\",\"pages\":\"365–377\",\"issn\":\"1565-8511\",\"urljournal\":\"https://doi.org/10.1007/s11856-018-1735-8\",\"abstract\":\"Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable $Π_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fraïsse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.\",\"bibtex\":\"@article{1701.07928,\\n   author = {I. Goldbring and B. Hart and H. Towsner},\\n    title = {Explicit sentences distinguishing McDuff's II$\\\\_1$ factors},\\n    urlarxiv={https://arxiv.org/abs/1701.07928},\\n    journal={Israel Journal of Mathematics},\\n doi = {10.1007/s11856-018-1735-8},\\nyear=2018,\\nmonth={Aug},\\nday=01,\\nvolume=227,\\nnumber=1,\\npages={365--377},\\nissn={1565-8511},\\nurljournal={https://doi.org/10.1007/s11856-018-1735-8},\\n    abstract={Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable $\\\\Pi_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\\\\\\\"isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.}\\n}\\n\\n\",\"author_short\":[\"Goldbring, I.\",\"Hart, B.\",\"Towsner, H.\"],\"key\":\"1701.07928\",\"id\":\"1701.07928\",\"bibbaseid\":\"goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"https://arxiv.org/abs/1701.07928\",\"Journal\":\"https://doi.org/10.1007/s11856-018-1735-8\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"William\"],\"propositions\":[],\"lastnames\":[\"Simmons\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Proof mining and effective bounds in differential polynomial rings\",\"journal\":\"Advances in Mathematics\",\"volume\":\"343\",\"pages\":\"567 - 623\",\"year\":\"2019\",\"issn\":\"0001-8708\",\"doi\":\"https://doi.org/10.1016/j.aim.2018.11.026\",\"urljournal\":\"http://www.sciencedirect.com/science/article/pii/S0001870818304900\",\"urlarxiv\":\"http://arxiv.org/abs/1609.07509\",\"abstract\":\"Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.\",\"bibtex\":\"@article{1609.07509,\\nauthor={William Simmons and Henry Towsner},\\ntitle={Proof mining and effective bounds in differential polynomial rings},\\njournal = {Advances in Mathematics},\\nvolume = 343,\\npages = {567 - 623},\\nyear = 2019,\\nissn = {0001-8708},\\ndoi = {https://doi.org/10.1016/j.aim.2018.11.026},\\nurljournal = {http://www.sciencedirect.com/science/article/pii/S0001870818304900},\\nurlarxiv={http://arxiv.org/abs/1609.07509},\\nabstract={Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.}\\n}\\n\\n\",\"author_short\":[\"Simmons, W.\",\"Towsner, H.\"],\"key\":\"1609.07509\",\"id\":\"1609.07509\",\"bibbaseid\":\"simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://www.sciencedirect.com/science/article/pii/S0001870818304900\",\"Arxiv\":\"http://arxiv.org/abs/1609.07509\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":34,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Constructing Sequences One Step at a Time\",\"journal\":\"Journal of Mathematical Logic\",\"year\":\"2020\",\"urlarxiv\":\"http://arxiv.org/abs/1609.05509\",\"doi\":\"10.1142/S0219061320500178\",\"urljournal\":\"https:///dx.doi.org/10.1142/S0219061320500178 \",\"volume\":\"20\",\"issue\":\"3\",\"abstract\":\"We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle \\\"the product of well-quasi-orders is a well-quasi-order\\\" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.\",\"bibtex\":\"@article{1609.05509,\\nauthor = {Henry Towsner},\\ntitle = {Constructing Sequences One Step at a Time},\\njournal = {Journal of Mathematical Logic},\\nyear = {2020},\\nurlarxiv={http://arxiv.org/abs/1609.05509},\\ndoi={10.1142/S0219061320500178},\\nurljournal={ https:///dx.doi.org/10.1142/S0219061320500178 },\\nvolume=20,\\nissue=3,\\n\\nabstract = {We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle \\\"the product of well-quasi-orders is a well-quasi-order\\\" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.}\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"1609.05509\",\"id\":\"1609.05509\",\"bibbaseid\":\"towsner-constructingsequencesonestepatatime-2020\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1609.05509\",\"Journal\":\"https:///dx.doi.org/10.1142/S0219061320500178 \"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":31,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"Steven\"],\"propositions\":[],\"lastnames\":[\"Lindell\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]},{\"firstnames\":[\"Scott\"],\"propositions\":[],\"lastnames\":[\"Weinstein\"],\"suffixes\":[]}],\"title\":\"Hanf Locality and Invariant Elementary Definability\",\"year\":\"2016\",\"note\":\"submitted\",\"abstract\":\"We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical rela tions. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call such expansions presentations of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect to elementary, neighborhood-bounded presentations. The second is a non-uniform version of the Fagin- Stockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, local presentations.\",\"bibtex\":\"@misc{hanf_locality,\\nauthor = {Steven Lindell and Henry Towsner and Scott Weinstein},\\ntitle = {Hanf Locality and Invariant Elementary Definability},\\nyear = {2016},\\nnote = {submitted},\\nabstract = {We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical rela tions. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call\\nsuch expansions presentations of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect\\nto elementary, neighborhood-bounded presentations. The second is a non-uniform version of the Fagin-\\nStockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, local presentations.}\\n}\\n\\n\",\"author_short\":[\"Lindell, S.\",\"Towsner, H.\",\"Weinstein, S.\"],\"key\":\"hanf_locality\",\"id\":\"hanf_locality\",\"bibbaseid\":\"lindell-towsner-weinstein-hanflocalityandinvariantelementarydefinability-2016\",\"role\":\"author\",\"urls\":{},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Harry\"],\"propositions\":[],\"lastnames\":[\"Crane\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Relative exchangeability with equivalence relations\",\"year\":\"2018\",\"volume\":\"57\",\"issue\":\"5\",\"pages\":\"533-556\",\"journal\":\"Archive for Mathematical Logic\",\"eprint\":\"arXiv:1605.04484\",\"doi\":\"10.1007/s00153-017-0591-2\",\"urlarxiv\":\"http://arxiv.org/abs/1605.04484\",\"urljournal\":\"http://link.springer.com/article/10.1007/s00153-017-0591-2\",\"abstract\":\"We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman i̧teAckerman2015 and Crane and Towsner i̧teCraneTowsner2015 and hierarchical exchangeability results due to Austin and Panchenko i̧teAustinPanchenko2014. \",\"bibtex\":\"@article{1605.04484,\\nAuthor = {Harry Crane and Henry Towsner},\\nTitle = {Relative exchangeability with equivalence relations},\\nYear = {2018},\\nvolume = {57},\\nissue = {5},\\npages={533-556},\\njournal={Archive for Mathematical Logic},\\nEprint = {arXiv:1605.04484},\\ndoi = {10.1007/s00153-017-0591-2},\\nurlarxiv={http://arxiv.org/abs/1605.04484},\\nurljournal={http://link.springer.com/article/10.1007/s00153-017-0591-2},\\nabstract={We describe an Aldous--Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \\\\cite{Ackerman2015} and Crane and Towsner \\\\cite{CraneTowsner2015} and hierarchical exchangeability results due to Austin and Panchenko \\\\cite{AustinPanchenko2014}.\\n}\\n}\\n\\n\",\"author_short\":[\"Crane, H.\",\"Towsner, H.\"],\"key\":\"1605.04484\",\"id\":\"1605.04484\",\"bibbaseid\":\"crane-towsner-relativeexchangeabilitywithequivalencerelations-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1605.04484\",\"Journal\":\"http://link.springer.com/article/10.1007/s00153-017-0591-2\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":11,\"html\":\"\"},{\"bibtype\":\"misc\",\"type\":\"misc\",\"author\":[{\"firstnames\":[\"Harry\"],\"propositions\":[],\"lastnames\":[\"Crane\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"The structure of combinatorial Markov processes\",\"year\":\"2016\",\"eprint\":\"arXiv:1603.05954\",\"note\":\"submitted\",\"urlarxiv\":\"http://arxiv.org/abs/1603.05954\",\"abstract\":\"Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the n-disjoint amalgamation property for all n >= 1, then its jump measure can be decomposed explicitly in the sense of Lévy–Itô– Khintchine.\",\"bibtex\":\"@misc{1603.05954,\\nAuthor = {Harry Crane and Henry Towsner},\\nTitle = {The structure of combinatorial Markov processes},\\nYear = {2016},\\nEprint = {arXiv:1603.05954},\\nnote={submitted},\\nurlarxiv={http://arxiv.org/abs/1603.05954},\\nabstract={ Every exchangeable Feller process taking values in a suitably\\nnice combinatorial state space can be constructed by a system of\\niterated random Lipschitz functions. In discrete time, the construction\\nproceeds by iterated application of independent, identically distributed\\nfunctions, while in continuous time the random functions occur as the\\natoms of a time homogeneous Poisson point process. We further show\\nthat every exchangeable Feller process projects to a Feller process in an\\nappropriate limit space, akin to the projection of partition-valued processes\\ninto the ranked-simplex and graph-valued processes into the space\\nof graph limits. Together, our main theorems establish common structural\\nfeatures shared by all exchangeable combinatorial Feller processes,\\nregardless of the dynamics or resident state space, thereby generalizing\\nbehaviors previously observed for exchangeable coalescent and fragmentation\\nprocesses as well as other combinatorial stochastic processes. If,\\nin addition, an exchangeable Feller process evolves on a state space satisfying\\nthe n-disjoint amalgamation property for all n >= 1, then its\\njump measure can be decomposed explicitly in the sense of Lévy–Itô–\\nKhintchine.}\\n}\\n\\n\\n\",\"author_short\":[\"Crane, H.\",\"Towsner, H.\"],\"key\":\"1603.05954\",\"id\":\"1603.05954\",\"bibbaseid\":\"crane-towsner-thestructureofcombinatorialmarkovprocesses-2016\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1603.05954\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Epsilon Substitution for $ID_1$ via Cut-Elimination\",\"year\":\"2018\",\"volume\":\"57\",\"issue\":\"5\",\"pages\":\"497-531\",\"journal\":\"Archive for Mathematical Logic\",\"doi\":\"10.1007/s00153-017-0590-3\",\"urlarxiv\":\"http://arxiv.org/abs/1509.00390\",\"urljournal\":\"https://link.springer.com/article/10.1007/s00153-017-0590-3\",\"abstract\":\"The epsilon-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory ID_1 using a variant of the cut-elimination formalism introduced by Mints.\",\"bibtex\":\"@ARTICLE{towsner15:epsilon_cutelim,\\nauthor = {{Towsner}, H.},\\ntitle=\\\"{Epsilon Substitution for $ID_1$ via Cut-Elimination}\\\",\\nyear=2018,\\nvolume = {57},\\nissue = {5},\\npages={497-531},\\njournal={Archive for Mathematical Logic},\\ndoi = {10.1007/s00153-017-0590-3},\\nurlarxiv={http://arxiv.org/abs/1509.00390},\\nurljournal={https://link.springer.com/article/10.1007/s00153-017-0590-3},\\nabstract={The epsilon-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory ID_1 using a variant of the cut-elimination formalism introduced by Mints.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner15:epsilon_cutelim\",\"id\":\"towsner15:epsilon_cutelim\",\"bibbaseid\":\"towsner-epsilonsubstitutionforid1viacutelimination-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1509.00390\",\"Journal\":\"https://link.springer.com/article/10.1007/s00153-017-0590-3\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":9,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Crane\"],\"firstnames\":[\"H.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Relatively exchangeable structures\",\"journal\":\"Journal of Symbolic Logic\",\"year\":\"2018\",\"volume\":\"83\",\"number\":\"2\",\"pages\":\"416-442\",\"urlarxiv\":\"http://arxiv.org/abs/1509.06733\",\"urljournal\":\"https://doi.org/10.1017/jsl.2017.61\",\"doi\":\"10.1017/jsl.2017.61\",\"abstract\":\"We study random relational structures that are \\\\emphrelatively exchangeable—that is, whose distributions are invariant under the automorphisms of a reference structure &#x1D510;. When &#x1D510; has <i>trivial definable closure</i>, every relatively exchangeable structure satisfies a general Aldous–Hoover-type representation. If &#x1D510; satisfies the stronger properties of <i>ultrahomogeneity</i> and <i>n-disjoint amalgamation property</i> (n-DAP) for every n&geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &#x1D510;.\",\"bibtex\":\"@ARTICLE{2015arXiv150906733C,\\n   author = {{Crane}, H. and {Towsner}, H.},\\n    title = \\\"{Relatively exchangeable structures}\\\",\\njournal={Journal of Symbolic Logic},\\nyear = 2018,\\nvolume={83},\\nnumber={2},\\npages={416-442},\\nurlarxiv={http://arxiv.org/abs/1509.06733},\\nurljournal={https://doi.org/10.1017/jsl.2017.61},\\ndoi={10.1017/jsl.2017.61},\\nabstract={We study random relational structures that are \\\\emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure &#x1D510;. When &#x1D510; has <i>trivial definable closure</i>, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If &#x1D510; satisfies the stronger properties of <i>ultrahomogeneity</i> and <i>n-disjoint amalgamation property</i> (n-DAP) for every n&geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &#x1D510;.},\\n}\\n\\n\\n\",\"author_short\":[\"Crane, H.\",\"Towsner, H.\"],\"key\":\"2015arXiv150906733C\",\"id\":\"2015arXiv150906733C\",\"bibbaseid\":\"crane-towsner-relativelyexchangeablestructures-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1509.06733\",\"Journal\":\"https://doi.org/10.1017/jsl.2017.61\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":10,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Igusa\"],\"firstnames\":[\"G.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets\",\"year\":\"2017\",\"volume\":\"56\",\"number\":\"1\",\"journal\":\"Archive for Mathematical Logic\",\"doi\":\"10.1007/s00153-016-0519-2\",\"urljournal\":\"http://link.springer.com/article/10.1007/s00153-016-0519-2\",\"urlarxiv\":\"http://arxiv.org/abs/1507.03256\",\"abstract\":\"In [J], Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite &Pi;<sup>0</sup><sup>2</sub> homogeneous set. The proof uses a countable collection of &Pi;<sup>0</sup><sup>2</sub> sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of &Pi;<sup>0</sup><sup>2</sub> sets. We provide a proof of this latter fact. \",\"bibtex\":\"@ARTICLE{igusa15:computable_ramseys_theorem,\\n   author = {{Igusa}, G. and {Towsner}, H.},\\n    title = \\\"{Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets}\\\",\\nyear=\\\"2017\\\",\\nvolume=\\\"56\\\",\\nnumber=\\\"1\\\",\\n    journal = {Archive for Mathematical Logic},\\n    doi=\\\"10.1007/s00153-016-0519-2\\\",\\n    urljournal={http://link.springer.com/article/10.1007/s00153-016-0519-2},\\nurlarxiv={http://arxiv.org/abs/1507.03256},\\nabstract={In [J], Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite &Pi;<sup>0</sup><sup>2</sub> homogeneous set. The proof uses a countable collection of &Pi;<sup>0</sup><sup>2</sub> sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of &Pi;<sup>0</sup><sup>2</sub> sets. We provide a proof of this latter fact.\\n}\\n}\\n\\n\\n\",\"author_short\":[\"Igusa, G.\",\"Towsner, H.\"],\"key\":\"igusa15:computable_ramseys_theorem\",\"id\":\"igusa15:computable_ramseys_theorem\",\"bibbaseid\":\"igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://link.springer.com/article/10.1007/s00153-016-0519-2\",\"Arxiv\":\"http://arxiv.org/abs/1507.03256\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"inbook\",\"type\":\"inbook\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"chapter\":\"8\",\"editor\":[{\"propositions\":[],\"lastnames\":[\"Patrick\",\"Cégielski\"],\"firstnames\":[\"Ali\",\"Enayat\"],\"suffixes\":[\"\"]},{\"firstnames\":[\"Roman\"],\"propositions\":[],\"lastnames\":[\"Kossak\"],\"suffixes\":[]}],\"publisher\":\"CSLI\",\"series\":\"Studies in Weak Arithmetic\",\"title\":\"Towards an Effective Theory of Absolutely Continuous Measures\",\"volume\":\"3\",\"year\":\"2016\",\"urljournal\":\"https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml\",\"urlarxiv\":\"http://arxiv.org/abs/1503.04743\",\"abstract\":\"We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence L^1 functions. A crucial tool is a one-dimensional version of Szemeredi's regularity lemma for L^1 functions. \",\"bibtex\":\"@inbook{towsner:abs_cont,\\n\\tAuthor = {Towsner, Henry},\\n\\tChapter = {8},\\n\\tEditor = {Patrick C\\\\'egielski, Ali Enayat, and Roman Kossak},\\n\\tPublisher = {CSLI},\\n\\tSeries = {Studies in Weak Arithmetic},\\n\\tTitle = {Towards an Effective Theory of Absolutely Continuous Measures},\\n\\tVolume = {3},\\n\\tYear = {2016},\\nurljournal={https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml},\\nurlarxiv={http://arxiv.org/abs/1503.04743},\\n    abstract={We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence L^1 functions. A crucial tool is a one-dimensional version of Szemeredi's regularity lemma for L^1 functions.\\n}\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"editor_short\":[\"Patrick Cégielski, A. E.\",\"Kossak, R.\"],\"key\":\"towsner:abs_cont\",\"id\":\"towsner:abs_cont\",\"bibbaseid\":\"towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016\",\"role\":\"author\",\"urls\":{\"Journal\":\"https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml\",\"Arxiv\":\"http://arxiv.org/abs/1503.04743\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":8,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"year\":\"2015\",\"issn\":\"0933-5846\",\"journal\":\"Archive for Mathematical Logic\",\"volume\":\"54\",\"number\":\"7-8\",\"doi\":\"10.1007/s00153-015-0449-4\",\"title\":\"Dividing and weak quasi-dimensions in arbitrary theories\",\"url\":\"http://dx.doi.org/10.1007/s00153-015-0449-4\",\"publisher\":\"Springer Berlin Heidelberg\",\"keywords\":\"Pseudofinite structure; Quasidimension; Dividing; 03C20; 03C45\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Goldbring\"],\"firstnames\":[\"Isaac\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"pages\":\"915-920\",\"language\":\"English\",\"abstract\":\"We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasidimension that detects dividing.\",\"bibtex\":\"@article{goldbring15:dividing,\\nyear={2015},\\nissn={0933-5846},\\njournal={Archive for Mathematical Logic},\\nvolume={54},\\nnumber={7-8},\\ndoi={10.1007/s00153-015-0449-4},\\ntitle={Dividing and weak quasi-dimensions in arbitrary theories},\\nurl={http://dx.doi.org/10.1007/s00153-015-0449-4},\\npublisher={Springer Berlin Heidelberg},\\nkeywords={Pseudofinite structure; Quasidimension; Dividing; 03C20; 03C45},\\nauthor={Goldbring, Isaac and Towsner, Henry},\\npages={915-920},\\nlanguage={English},\\nabstract={We show that any countable model of a model complete\\ntheory has an elementary extension with a “pseudofinite-like” quasidimension\\nthat detects dividing.}\\n}\\n\\n\\n\",\"author_short\":[\"Goldbring, I.\",\"Towsner, H.\"],\"key\":\"goldbring15:dividing\",\"id\":\"goldbring15:dividing\",\"bibbaseid\":\"goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015\",\"role\":\"author\",\"urls\":{\"Paper\":\"http://dx.doi.org/10.1007/s00153-015-0449-4\"},\"keyword\":[\"Pseudofinite structure; Quasidimension; Dividing; 03C20; 03C45\"],\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space\",\"journal\":\"Houston Journal of Mathematics\",\"year\":\"2018\",\"month\":\"August\",\"volume\":\"44\",\"issue\":\"3\",\"pages\":\"873–885\",\"urljournal\":\"https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf\",\"urlarxiv\":\"http://arxiv.org/abs/1503.04745\",\"abstract\":\"The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.\",\"bibtex\":\"@article{towsner15:lower_bound_james_space,\\n   author = {{Towsner}, H.},\\n    title = \\\"{An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space}\\\",\\njournal={Houston Journal of Mathematics},\\n     year = 2018,\\n    month = aug,\\n    volume = 44,\\n    issue = 3,\\n    pages = {873--885},\\n    urljournal={https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf},\\n    urlarxiv={http://arxiv.org/abs/1503.04745},\\n    abstract={The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner15:lower_bound_james_space\",\"id\":\"towsner15:lower_bound_james_space\",\"bibbaseid\":\"towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018\",\"role\":\"author\",\"urls\":{\"Journal\":\"https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf\",\"Arxiv\":\"http://arxiv.org/abs/1503.04745\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":7,\"html\":\"\"},{\"bibtype\":\"unpublished\",\"type\":\"unpublished\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"A Worked Example of the Functional Interpretation\",\"note\":\"submitted\",\"year\":\"2015\",\"month\":\"March\",\"urlarxiv\":\"http://arxiv.org/abs/1503.05572\",\"abstract\":\"The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.\",\"bibtex\":\"@unpublished{towsner15:worked_example,\\n   author = {{Towsner}, H.},\\n    title = \\\"{A Worked Example of the Functional Interpretation}\\\",\\n    note={submitted},\\n     year = 2015,\\n    month = mar,\\n   urlarxiv = {http://arxiv.org/abs/1503.05572},\\n  abstract = {The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner15:worked_example\",\"id\":\"towsner15:worked_example\",\"bibbaseid\":\"towsner-aworkedexampleofthefunctionalinterpretation-2015\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1503.05572\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":10,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Flood\"],\"firstnames\":[\"S.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Separating Principles Below WKL0\",\"journal\":\"Mathematical Logic Quarterly\",\"year\":\"2016\",\"month\":\"December\",\"volume\":\"62\",\"number\":\"6\",\"issn\":\"1521-3870\",\"urljournal\":\"http://dx.doi.org/10.1002/malq.201500001\",\"doi\":\"10.1002/malq.201500001\",\"pages\":\"507–529\",\"urlarxiv\":\"http://arxiv.org/abs/1410.4068\",\"abstract\":\"In this paper, we study Ramsey-type Konig's Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a \\\"one step\\\" construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). We illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL from DNR (which has been shown separately by Bienvenu, Patey, and Schafer). \",\"bibtex\":\"@article{flood14:separating_principles_below_wkl0,\\n   author = {{Flood}, S. and {Towsner}, H.},\\n    title = \\\"{Separating Principles Below WKL0}\\\",\\njournal={Mathematical Logic Quarterly},\\nyear={2016},\\nmonth=dec,\\nvolume = {62},\\nnumber = {6},\\nissn = {1521-3870},\\nurljournal = {http://dx.doi.org/10.1002/malq.201500001},\\ndoi = {10.1002/malq.201500001},\\npages = {507--529},\\nurlarxiv={http://arxiv.org/abs/1410.4068},\\nabstract={In this paper, we study Ramsey-type Konig's Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a \\\"one step\\\" construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). \\n\\nWe illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL from DNR (which has been shown separately by Bienvenu, Patey, and Schafer).\\n}\\n}\\n\\n\\n\\n\\n\",\"author_short\":[\"Flood, S.\",\"Towsner, H.\"],\"key\":\"flood14:separating_principles_below_wkl0\",\"id\":\"flood14:separating_principles_below_wkl0\",\"bibbaseid\":\"flood-towsner-separatingprinciplesbelowwkl0-2016\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1002/malq.201500001\",\"Arxiv\":\"http://arxiv.org/abs/1410.4068\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]}],\"title\":\"Limits of sequences of Markov chains\",\"journal\":\"Electron. J. Probab.\",\"fjournal\":\"Electronic Journal of Probability\",\"volume\":\"20\",\"year\":\"2015\",\"keywords\":\"Markov chain, graph limit, ultraproduct\",\"abstract\":\"We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called \\\"random-free\\\" or \\\"product measurable\\\") is simpler than the general case.\",\"pages\":\"no. 77, 1-23\",\"issn\":\"1083-6489\",\"doi\":\"10.1214/EJP.v20-4188\",\"urljournal\":\"http://ejp.ejpecp.org/article/view/4188\",\"urlarxiv\":\"http://arxiv.org/abs/1404.3815\",\"bibtex\":\"@article{towsner14:sequence_markov_chains,\\n\\tauthor = {Henry Towsner},\\n\\ttitle = {Limits of sequences of Markov chains},\\n\\tjournal = {Electron. J. Probab.},\\n\\tfjournal = {Electronic Journal of Probability},\\n\\tvolume = {20},\\n\\tyear = {2015},\\n\\tkeywords = {Markov chain, graph limit, ultraproduct},\\n\\tabstract = {We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space.  The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called \\\"random-free\\\" or \\\"product measurable\\\") is simpler than the general case.},\\n\\tpages = {no. 77, 1-23},\\n\\tissn = {1083-6489},\\n\\tdoi = {10.1214/EJP.v20-4188},    \\n        urljournal = {http://ejp.ejpecp.org/article/view/4188},\\nurlarxiv={http://arxiv.org/abs/1404.3815},\\n}\\n\\n\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner14:sequence_markov_chains\",\"id\":\"towsner14:sequence_markov_chains\",\"bibbaseid\":\"towsner-limitsofsequencesofmarkovchains-2015\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://ejp.ejpecp.org/article/view/4188\",\"Arxiv\":\"http://arxiv.org/abs/1404.3815\"},\"keyword\":[\"Markov chain\",\"graph limit\",\"ultraproduct\"],\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":8,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Sigma-Algebras for Quasirandom Hypergraphs\",\"journal\":\"Random Struct. Alg.\",\"fjournal\":\"Random Structures & Algoritmhs\",\"volume\":\"50\",\"number\":\"1\",\"pages\":\"114–139\",\"year\":\"2017\",\"urljournal\":\"http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract\",\"doi\":\"10.1002/rsa.20641\",\"urlarxiv\":\"http://arxiv.org/abs/1312.4882\",\"abstract\":\"We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. We associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.\",\"bibtex\":\"@article{towsner13:quasirandom_hypergraphs,\\nauthor = {{Towsner}, H.},\\ntitle = {Sigma-Algebras for Quasirandom Hypergraphs},\\njournal = {Random Struct. Alg.},\\nfjournal = {Random Structures \\\\& Algoritmhs},\\nvolume = {50},\\nnumber = {1},\\npages = {114--139},\\nyear = {2017},\\nurljournal={http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract},\\ndoi={10.1002/rsa.20641},\\nurlarxiv={http://arxiv.org/abs/1312.4882},\\nabstract = {We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. \\nWe associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, R\\\\\\\"odl, and Schacht; Conlon, H\\\\`an, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.}\\n}\\n\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner13:quasirandom_hypergraphs\",\"id\":\"towsner13:quasirandom_hypergraphs\",\"bibbaseid\":\"towsner-sigmaalgebrasforquasirandomhypergraphs-2017\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract\",\"Arxiv\":\"http://arxiv.org/abs/1312.4882\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":11,\"html\":\"\"},{\"bibtype\":\"incollection\",\"type\":\"incollection\",\"author\":[{\"firstnames\":[\"Steven\"],\"propositions\":[],\"lastnames\":[\"Lindell\"],\"suffixes\":[]},{\"firstnames\":[\"Henry\"],\"propositions\":[],\"lastnames\":[\"Towsner\"],\"suffixes\":[]},{\"firstnames\":[\"Scott\"],\"propositions\":[],\"lastnames\":[\"Weinstein\"],\"suffixes\":[]}],\"title\":\"Infinitary methods in finite model theory\",\"booktitle\":\"Logic Without Borders\",\"publisher\":\"de Gruyter\",\"year\":\"2015\",\"editor\":[{\"firstnames\":[\"Asa\"],\"propositions\":[],\"lastnames\":[\"Hirvonen\"],\"suffixes\":[]},{\"firstnames\":[\"Juha\"],\"propositions\":[],\"lastnames\":[\"Kontinen\"],\"suffixes\":[]},{\"firstnames\":[\"Roman\"],\"propositions\":[],\"lastnames\":[\"Kossak\"],\"suffixes\":[]},{\"firstnames\":[\"Andrés\"],\"propositions\":[],\"lastnames\":[\"Villaveces\"],\"suffixes\":[]}],\"urlpdf\":\"http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf\",\"abstract\":\"This paper explores applications of the infinitary techniques of classical model theory in the context of finite model theory, and thus elaborates on themes explored by V&auml;&auml;n&auml;nen (2003). We are grateful for the opportunity to celebrate Jouko's inclusive vision of logic in this way. \",\"bibtex\":\"@InCollection{lindell13:_infin,\\n  author = \\t {Steven Lindell and Henry Towsner and Scott Weinstein},\\n  title = \\t {Infinitary methods in finite model theory},\\n  booktitle = {Logic Without Borders},\\n  publisher = {de Gruyter},\\n  year = \\t 2015,\\n  editor = \\t {Asa Hirvonen and Juha Kontinen and Roman Kossak and Andr\\\\'es Villaveces},\\n  urlpdf = {http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf},\\nabstract={\\nThis paper explores applications of the infinitary techniques of classical model theory in\\nthe context of finite model theory, and thus elaborates on themes explored by V&auml;&auml;n&auml;nen\\n(2003). We are grateful for the opportunity to celebrate Jouko's inclusive vision of logic\\nin this way.\\n}\\n}\\n\\n\",\"author_short\":[\"Lindell, S.\",\"Towsner, H.\",\"Weinstein, S.\"],\"editor_short\":[\"Hirvonen, A.\",\"Kontinen, J.\",\"Kossak, R.\",\"Villaveces, A.\"],\"key\":\"lindell13:_infin\",\"id\":\"lindell13:_infin\",\"bibbaseid\":\"lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015\",\"role\":\"author\",\"urls\":{\"Pdf\":\"http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"On Maximum Conservative Extensions\",\"journal\":\"Computability\",\"year\":\"2015\",\"volume\":\"4\",\"pages\":\"57–68\",\"urlarxiv\":\"http://arxiv.org/abs/1302.1488\",\"abstract\":\"We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma-n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi-2 complete.\",\"bibtex\":\"@Article{towsner13:maximal_conservatve,\\n  author = \\t {Towsner, H.},\\n  title = \\t {On Maximum Conservative Extensions},\\n  journal = \\t {Computability},\\n  year = \\t 2015,\\n  volume = \\t 4,\\n  pages = \\t {57--68},\\nurlarxiv={http://arxiv.org/abs/1302.1488},\\nabstract={We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma-n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi-2 complete.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner13:maximal_conservatve\",\"id\":\"towsner13:maximal_conservatve\",\"bibbaseid\":\"towsner-onmaximumconservativeextensions-2015\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1302.1488\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":12,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Lerman\"],\"firstnames\":[\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Solomon\"],\"firstnames\":[\"R.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"H.\"],\"suffixes\":[]}],\"title\":\"Separating principles below Ramsey's Theorem for Pairs\",\"journal\":\"J. Math. Logic\",\"fjournal\":\"Journal of Mathematical Logic\",\"volume\":\"13\",\"number\":\"02\",\"pages\":\"1350007\",\"year\":\"2013\",\"doi\":\"10.1142/S0219061313500074\",\"urljournal\":\"http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074\",\"urlarxiv\":\"http://arxiv.org/abs/1302.0828\",\"abstract\":\"In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the \\\"big five\\\" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.\",\"bibtex\":\"@article{doi:10.1142/S0219061313500074,\\nauthor = {{Lerman}, M. and {Solomon}, R. and {Towsner}, H.},\\ntitle = {Separating principles below Ramsey's Theorem for Pairs},\\njournal={J. Math. Logic},\\nfjournal={Journal of Mathematical Logic},\\nvolume = {13},\\nnumber = {02},\\npages = {1350007},\\nyear = {2013},\\ndoi = {10.1142/S0219061313500074},\\nURLjournal = {http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074},\\nurlarxiv={http://arxiv.org/abs/1302.0828},\\nabstract={In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the \\\"big five\\\" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.},\\n}\\n\\n\\n\\n\",\"author_short\":[\"Lerman, M.\",\"Solomon, R.\",\"Towsner, H.\"],\"key\":\"doi:10.1142/S0219061313500074\",\"id\":\"doi:10.1142/S0219061313500074\",\"bibbaseid\":\"lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074\",\"Arxiv\":\"http://arxiv.org/abs/1302.0828\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":35,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Franklin\"],\"firstnames\":[\"Johanna\",\"N.\",\"Y.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Randomness and non-ergodic systems\",\"journal\":\"Mosc. Math. J.\",\"fjournal\":\"Moscow Mathematical Journal\",\"volume\":\"14\",\"year\":\"2014\",\"number\":\"4\",\"pages\":\"711–744, 827\",\"issn\":\"1609-3321\",\"mrclass\":\"37A25 (03D32)\",\"mrnumber\":\"3292047\",\"urlarxiv\":\"http://arxiv.org/abs/1206.2682\",\"urljournal\":\"http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html\",\"abstract\":\"We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.\",\"bibtex\":\"@article {franklin12:_random_non_system,\\n    AUTHOR = {Franklin, Johanna N. Y. and Towsner, Henry},\\n     TITLE = {Randomness and non-ergodic systems},\\n   JOURNAL = {Mosc. Math. J.},\\n  FJOURNAL = {Moscow Mathematical Journal},\\n    VOLUME = {14},\\n      YEAR = {2014},\\n    NUMBER = {4},\\n     PAGES = {711--744, 827},\\n      ISSN = {1609-3321},\\n   MRCLASS = {37A25 (03D32)},\\n  MRNUMBER = {3292047},\\nurlarxiv={http://arxiv.org/abs/1206.2682},\\nurljournal={http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html},\\nabstract={We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.},\\n}\\n\\n\\n\",\"author_short\":[\"Franklin, J. N. Y.\",\"Towsner, H.\"],\"key\":\"franklin12:_random_non_system\",\"id\":\"franklin12:_random_non_system\",\"bibbaseid\":\"franklin-towsner-randomnessandnonergodicsystems-2014\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1206.2682\",\"Journal\":\"http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal\",\"year\":\"2018\",\"journal\":\"Discrete Analysis\",\"number\":\"4\",\"urlarxiv\":\"http://arxiv.org/abs/1204.1884\",\"urljournal\":\"http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal\",\"abstract\":\"The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.\",\"bibtex\":\"@article{towsner12:_analy_approac_spars_hyper,\\n  author = \\t {Towsner, Henry},\\n  title = \\t {An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal},\\n  year = \\t 2018,\\n  journal={Discrete Analysis},\\n  number={4},\\nurlarxiv={http://arxiv.org/abs/1204.1884},\\nurljournal={http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal},\\nabstract={The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.},\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner12:_analy_approac_spars_hyper\",\"id\":\"towsner12:_analy_approac_spars_hyper\",\"bibbaseid\":\"towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1204.1884\",\"Journal\":\"http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":16,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Ultrafilters in reverse mathematics\",\"journal\":\"J. Math. Logic\",\"fjournal\":\"Journal of Mathematical Logic\",\"volume\":\"14\",\"number\":\"01\",\"pages\":\"1450001\",\"year\":\"2014\",\"doi\":\"10.1142/S0219061314500019\",\"urlarxiv\":\"http://arxiv.org/abs/1109.3902\",\"abstract\":\"We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA<sub>0</sub>, ATR<sub>0</sub>, and &Pi;<sup>1</sup><sub>1</sub>-Comprehension.\",\"urljournal\":\"http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019\",\"bibtex\":\"@article{doi:10.1142/S0219061314500019,\\nauthor = {Towsner, Henry},\\ntitle = {Ultrafilters in reverse mathematics},\\njournal={J. Math. Logic},\\nfjournal={Journal of Mathematical Logic},\\nvolume = {14},\\nnumber = {01},\\npages = {1450001},\\nyear = {2014},\\ndoi = {10.1142/S0219061314500019},\\nurlarxiv={http://arxiv.org/abs/1109.3902},\\nabstract={We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA<sub>0</sub>, ATR<sub>0</sub>, and &Pi;<sup>1</sup><sub>1</sub>-Comprehension.},\\nurljournal = {http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019}\\n}\\n\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"doi:10.1142/S0219061314500019\",\"id\":\"doi:10.1142/S0219061314500019\",\"bibbaseid\":\"towsner-ultrafiltersinreversemathematics-2014\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/1109.3902\",\"Journal\":\"http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":155,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Partial Impredicativity in Reverse Mathematics\",\"journal\":\"J. Symbolic Logic\",\"fjournal\":\"Journal of Symbolic Logic\",\"volume\":\"78\",\"number\":\"2\",\"pages\":\"459–488\",\"urljournal\":\"http://projecteuclid.org/euclid.jsl/1368627060\",\"year\":\"2013\",\"urlarxiv\":\"http://arxiv.org/abs/1106.6063\",\"abstract\":\"In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a &Pi;<sup>1</sup><sub>2</sub> sentence from the theory &Pi;<sup>1</sup><sub>1</sub>-Comprehension. Using methods based on the functional interpretation, we introduce a family of weakenings of &Pi;<sup>1</sup><sub>1</sub>-Comprehension and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.\",\"bibtex\":\"@article{towsner11:_approx_impred_rever_mathem,\\n  author = \\t {Towsner, Henry},\\n  title = \\t {Partial Impredicativity in Reverse Mathematics},\\njournal={J. Symbolic Logic},\\nfjournal={Journal of Symbolic Logic},\\nvolume={78},\\nnumber={2},\\npages={459--488},\\nurljournal={http://projecteuclid.org/euclid.jsl/1368627060},\\n  year = \\t 2013,\\nurlarxiv={http://arxiv.org/abs/1106.6063},\\nabstract={In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a &Pi;<sup>1</sup><sub>2</sub> sentence from the theory &Pi;<sup>1</sup><sub>1</sub>-Comprehension. Using methods based on the functional interpretation, we introduce a family of weakenings of &Pi;<sup>1</sup><sub>1</sub>-Comprehension and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.}\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner11:_approx_impred_rever_mathem\",\"id\":\"towsner11:_approx_impred_rever_mathem\",\"bibbaseid\":\"towsner-partialimpredicativityinreversemathematics-2013\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://projecteuclid.org/euclid.jsl/1368627060\",\"Arxiv\":\"http://arxiv.org/abs/1106.6063\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":33,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"year\":\"2014\",\"issn\":\"0021-2172\",\"journal\":\"Israel J. Math.\",\"fjournal\":\"Israel Journal of Mathematics\",\"volume\":\"199\",\"number\":\"2\",\"doi\":\"10.1007/s11856-013-0054-3\",\"title\":\"An approximate logic for measures\",\"urljournal\":\"http://dx.doi.org/10.1007/s11856-013-0054-3\",\"publisher\":\"The Hebrew University Magnes Press\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Goldbring\"],\"firstnames\":[\"Isaac\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"pages\":\"867-913\",\"language\":\"English\",\"urlarxiv\":\"http://arxiv.org/abs/1106.2854\",\"abstract\":\"We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer&eacute;di Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.\",\"bibtex\":\"@article{goldbring:_approx_logic_measur,\\nyear={2014},\\nissn={0021-2172},\\njournal={Israel J. Math.},\\nfjournal={Israel Journal of Mathematics},\\nvolume={199},\\nnumber={2},\\ndoi={10.1007/s11856-013-0054-3},\\ntitle={An approximate logic for measures},\\nurljournal={http://dx.doi.org/10.1007/s11856-013-0054-3},\\npublisher={The Hebrew University Magnes Press},\\nauthor={Goldbring, Isaac and Towsner, Henry},\\npages={867-913},\\nlanguage={English},\\nurlarxiv={http://arxiv.org/abs/1106.2854},\\nabstract={We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer&eacute;di Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.},\\n}\\n\\n\\n\",\"author_short\":[\"Goldbring, I.\",\"Towsner, H.\"],\"key\":\"goldbring:_approx_logic_measur\",\"id\":\"goldbring:_approx_logic_measur\",\"bibbaseid\":\"goldbring-towsner-anapproximatelogicformeasures-2014\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1007/s11856-013-0054-3\",\"Arxiv\":\"http://arxiv.org/abs/1106.2854\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":69,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"A combinatorial proof of the dense Hindman's theorem\",\"journal\":\"Discrete Math.\",\"fjournal\":\"Discrete Mathematics\",\"volume\":\"311\",\"year\":\"2011\",\"number\":\"14\",\"pages\":\"1380–1384\",\"issn\":\"0012-365X\",\"coden\":\"DSMHA4\",\"mrclass\":\"05D10 (03C20)\",\"mrnumber\":\"2795548 (2012h:05349)\",\"mrreviewer\":\"Damir D. Dzhafarov\",\"doi\":\"10.1016/j.disc.2011.03.006\",\"urljournal\":\"http://dx.doi.org/10.1016/j.disc.2011.03.006\",\"urlarxiv\":\"http://arxiv.org/abs/1002.0347\",\"abstract\":\"The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a \\\"dense\\\" set B<sub>1</sub>, for each b<sub>1</sub>&isin; B<sub>1</sub> a \\\"dense\\\" set B<sub>2</sub>(b<sub>1</sub>) (depending on b<sub>1</sub>), for each b<sub>2</sub>&isin; B<sub>2</sub>(b<sub>1</sub>) a \\\"dense\\\" set B<sub>3</sub>(b<sub>1</sub>, b<sub>2</sub>) (depending on b<sub>1</sub>, b<sub>2</sub>), and so on, such that for any such sequence of b<sub>i</sub>, all finite sums belong to the chosen color. (Here density is often taken to be \\\"piecewise syndetic\\\", but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.\",\"bibtex\":\"@article {MR2795548,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {A combinatorial proof of the dense {H}indman's theorem},\\n   JOURNAL = {Discrete Math.},\\n  FJOURNAL = {Discrete Mathematics},\\n    VOLUME = {311},\\n      YEAR = {2011},\\n    NUMBER = {14},\\n     PAGES = {1380--1384},\\n      ISSN = {0012-365X},\\n     CODEN = {DSMHA4},\\n   MRCLASS = {05D10 (03C20)},\\n  MRNUMBER = {2795548 (2012h:05349)},\\nMRREVIEWER = {Damir D. Dzhafarov},\\n       DOI = {10.1016/j.disc.2011.03.006},\\n       URLJOURNAL= {http://dx.doi.org/10.1016/j.disc.2011.03.006},\\nurlarxiv={http://arxiv.org/abs/1002.0347},\\nabstract={The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a \\\"dense\\\" set B<sub>1</sub>, for each b<sub>1</sub>&isin; B<sub>1</sub> a \\\"dense\\\" set B<sub>2</sub>(b<sub>1</sub>) (depending on b<sub>1</sub>), for each b<sub>2</sub>&isin; B<sub>2</sub>(b<sub>1</sub>) a \\\"dense\\\" set B<sub>3</sub>(b<sub>1</sub>, b<sub>2</sub>) (depending on b<sub>1</sub>, b<sub>2</sub>), and so on, such that for any such sequence of b<sub>i</sub>, all finite sums belong to the chosen color. (Here density is often taken to be \\\"piecewise syndetic\\\", but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.}\\n}\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2795548\",\"id\":\"MR2795548\",\"bibbaseid\":\"towsner-acombinatorialproofofthedensehindmanstheorem-2011\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1016/j.disc.2011.03.006\",\"Arxiv\":\"http://arxiv.org/abs/1002.0347\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":44,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Beiglbock\"],\"firstnames\":[\"Mathias\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Transfinite approximation of Hindman's theorem\",\"journal\":\"Israel J. Math.\",\"fjournal\":\"Israel Journal of Mathematics\",\"volume\":\"191\",\"year\":\"2012\",\"pages\":\"41–59\",\"issn\":\"0021-2172\",\"coden\":\"ISJMAP\",\"mrclass\":\"Preliminary Data\",\"mrnumber\":\"2970862\",\"doi\":\"10.1007/s11856-011-0195-1\",\"urljournal\":\"http://dx.doi.org/10.1007/s11856-011-0195-1\",\"urlarxiv\":\"http://arxiv.org/abs/1001.1175\",\"abstract\":\"Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a \\\"transfinite\\\" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.\",\"bibtex\":\"@article {MR2970862,\\n    AUTHOR = {Beiglbock, Mathias and Towsner, Henry},\\n     TITLE = {Transfinite approximation of {H}indman's theorem},\\n   JOURNAL = {Israel J. Math.},\\n  FJOURNAL = {Israel Journal of Mathematics},\\n    VOLUME = {191},\\n      YEAR = {2012},\\n     PAGES = {41--59},\\n      ISSN = {0021-2172},\\n     CODEN = {ISJMAP},\\n   MRCLASS = {Preliminary Data},\\n  MRNUMBER = {2970862},\\n       DOI = {10.1007/s11856-011-0195-1},\\n       URLJOURNAL = {http://dx.doi.org/10.1007/s11856-011-0195-1},\\nurlarxiv={http://arxiv.org/abs/1001.1175},\\nabstract={Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a \\\"transfinite\\\" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.},\\n}\\n\\n\",\"author_short\":[\"Beiglbock, M.\",\"Towsner, H.\"],\"key\":\"MR2970862\",\"id\":\"MR2970862\",\"bibbaseid\":\"beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1007/s11856-011-0195-1\",\"Arxiv\":\"http://arxiv.org/abs/1001.1175\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":45,\"html\":\"\"},{\"bibtype\":\"unpublished\",\"type\":\"unpublished\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Metastability in the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemerédi's Theorem\",\"note\":\"draft\",\"year\":\"2009\",\"urlarxiv\":\"http://arxiv.org/abs/0909.5668\",\"abstract\":\"The Furstenberg-Zimmer structure theorem for &integers;<sub>d,</sub> actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, &omega;<sup>&omega;<sup>&omega;<sup>&omega;</sup></sup></sup>). \",\"bibtex\":\"@Unpublished{towsner09:_metas_furst_zimmer_tower_ii,\\n  author = \\t {Towsner, Henry},\\n  title = \\t {Metastability in the {F}urstenberg-{Z}immer Tower II: Polynomial and Multidimensional {S}zemer\\\\'edi's Theorem},\\n  note = \\t {draft},\\n  year = \\t 2009,\\nurlarxiv={http://arxiv.org/abs/0909.5668},\\nabstract={The Furstenberg-Zimmer structure theorem for &integers;<sub>d,</sub> actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemer\\\\'edi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemer\\\\'edi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, &omega;<sup>&omega;<sup>&omega;<sup>&omega;</sup></sup></sup>).\\n}\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner09:_metas_furst_zimmer_tower_ii\",\"id\":\"towsner09:_metas_furst_zimmer_tower_ii\",\"bibbaseid\":\"towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/0909.5668\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"A Simple Proof and Some Difficult Examples for Hindman's Theorem\",\"journal\":\"Notre Dame J. Formal Logic\",\"fjournal\":\"Notre Dame Journal of Formal Logic\",\"year\":\"2012\",\"volume\":\"53\",\"number\":\"1\",\"pages\":\"53-65\",\"urljournal\":\"http://projecteuclid.org/euclid.ndjfl/1336586237\",\"urlarxiv\":\"http://arxiv.org/abs/0906.3885\",\"abstract\":\"We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.\",\"bibtex\":\"@Article{towsner12:_simpl_proof_some_diffic_examp_hindm_theor,\\n  author = \\t {Towsner, Henry},\\n  title = \\t {A Simple Proof and Some Difficult Examples for {H}indman's Theorem},\\njournal = {Notre Dame J. Formal Logic},\\n  fjournal = \\t {Notre Dame Journal of Formal Logic},\\n  year = \\t 2012,\\n  volume = \\t 53,\\n  number = \\t 1,\\n  pages = \\t {53-65},\\nurljournal={http://projecteuclid.org/euclid.ndjfl/1336586237},\\nurlarxiv={http://arxiv.org/abs/0906.3885},\\nabstract={We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner12:_simpl_proof_some_diffic_examp_hindm_theor\",\"id\":\"towsner12:_simpl_proof_some_diffic_examp_hindm_theor\",\"bibbaseid\":\"towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://projecteuclid.org/euclid.ndjfl/1336586237\",\"Arxiv\":\"http://arxiv.org/abs/0906.3885\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":47,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Hindman's theorem: an ultrafilter argument in second order arithmetic\",\"journal\":\"J. Symbolic Logic\",\"fjournal\":\"Journal of Symbolic Logic\",\"volume\":\"76\",\"year\":\"2011\",\"number\":\"1\",\"pages\":\"353–360\",\"issn\":\"0022-4812\",\"coden\":\"JSYLA6\",\"mrclass\":\"03F35 (54D80)\",\"mrnumber\":\"2791353 (2012b:03165)\",\"mrreviewer\":\"M. Yasuhara\",\"doi\":\"10.2178/jsl/1294171005\",\"urljournal\":\"http://dx.doi.org/10.2178/jsl/1294171005\",\"urlarxiv\":\"http://arxiv.org/abs/0906.3882\",\"abstract\":\"Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.\",\"bibtex\":\"@article {MR2791353,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {Hindman's theorem: an ultrafilter argument in second order\\n              arithmetic},\\n   JOURNAL = {J. Symbolic Logic},\\n  FJOURNAL = {Journal of Symbolic Logic},\\n    VOLUME = {76},\\n      YEAR = {2011},\\n    NUMBER = {1},\\n     PAGES = {353--360},\\n      ISSN = {0022-4812},\\n     CODEN = {JSYLA6},\\n   MRCLASS = {03F35 (54D80)},\\n  MRNUMBER = {2791353 (2012b:03165)},\\nMRREVIEWER = {M. Yasuhara},\\n       DOI = {10.2178/jsl/1294171005},\\n       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1294171005},\\nurlarxiv={http://arxiv.org/abs/0906.3882},\\nabstract={Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.},\\n}\\n\\n\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2791353\",\"id\":\"MR2791353\",\"bibbaseid\":\"towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.2178/jsl/1294171005\",\"Arxiv\":\"http://arxiv.org/abs/0906.3882\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":56,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"A Correspondence Principle for the Gowers Uniformity Norms\",\"journal\":\"Journal of Logic and Analysis\",\"year\":\"2012\",\"volume\":\"4\",\"month\":\"February\",\"urljournal\":\"http://logicandanalysis.org/index.php/jla/article/view/139/51\",\"urlarxiv\":\"http://arxiv.org/abs/0905.0493\",\"abstract\":\"We give a variation of the Furstenberg Correspondence which preserves the Gowers uniformity norms.\",\"bibtex\":\"@Article{towsner12:_corres_princ_gower_unifor_norms,\\n  author = \\t {Towsner, Henry},\\n  title = \\t {A Correspondence Principle for the {G}owers Uniformity Norms},\\n  journal = \\t {Journal of Logic and Analysis},\\n  year = \\t 2012,\\n  volume = \\t 4,\\n  month = \\t {February},\\nurljournal={http://logicandanalysis.org/index.php/jla/article/view/139/51},\\nurlarxiv={http://arxiv.org/abs/0905.0493},\\nabstract={We give a variation of the Furstenberg Correspondence which preserves the Gowers uniformity norms.},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"towsner12:_corres_princ_gower_unifor_norms\",\"id\":\"towsner12:_corres_princ_gower_unifor_norms\",\"bibbaseid\":\"towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://logicandanalysis.org/index.php/jla/article/view/139/51\",\"Arxiv\":\"http://arxiv.org/abs/0905.0493\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":30,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Avigad\"],\"firstnames\":[\"Jeremy\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Metastability in the Furstenberg-Zimmer tower\",\"journal\":\"Fund. Math.\",\"fjournal\":\"Fundamenta Mathematicae\",\"volume\":\"210\",\"year\":\"2010\",\"number\":\"3\",\"pages\":\"243–268\",\"issn\":\"0016-2736\",\"mrclass\":\"37A05 (03E15 03F03 28D05 37A25)\",\"mrnumber\":\"2733051 (2012b:37003)\",\"mrreviewer\":\"Michael H. Schraudner\",\"doi\":\"10.4064/fm210-3-2\",\"urljournal\":\"http://dx.doi.org/10.4064/fm210-3-2\",\"urlarxiv\":\"http://arxiv.org/abs/0902.0356\",\"abstract\":\"According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer&eacute;di's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the &ω;<sup>&ω;<sup>&ω;</sup></sup>th level.\",\"bibtex\":\"@article {MR2733051,\\n    AUTHOR = {Avigad, Jeremy and Towsner, Henry},\\n     TITLE = {Metastability in the {F}urstenberg-{Z}immer tower},\\n   JOURNAL = {Fund. Math.},\\n  FJOURNAL = {Fundamenta Mathematicae},\\n    VOLUME = {210},\\n      YEAR = {2010},\\n    NUMBER = {3},\\n     PAGES = {243--268},\\n      ISSN = {0016-2736},\\n   MRCLASS = {37A05 (03E15 03F03 28D05 37A25)},\\n  MRNUMBER = {2733051 (2012b:37003)},\\nMRREVIEWER = {Michael H. Schraudner},\\n       DOI = {10.4064/fm210-3-2},\\n       URLJOURNAL= {http://dx.doi.org/10.4064/fm210-3-2},\\nurlarxiv={http://arxiv.org/abs/0902.0356},\\nabstract={According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer&eacute;di's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the &\\\\omega;<sup>&\\\\omega;<sup>&\\\\omega;</sup></sup>th level.}\\n}\\n\\n\",\"author_short\":[\"Avigad, J.\",\"Towsner, H.\"],\"key\":\"MR2733051\",\"id\":\"MR2733051\",\"bibbaseid\":\"avigad-towsner-metastabilityinthefurstenbergzimmertower-2010\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.4064/fm210-3-2\",\"Arxiv\":\"http://arxiv.org/abs/0902.0356\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":23,\"html\":\"\"},{\"bibtype\":\"incollection\",\"type\":\"incollection\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Priority arguments and epsilon substitution\",\"booktitle\":\"Proofs, categories and computations\",\"series\":\"Tributes\",\"volume\":\"13\",\"pages\":\"251–266\",\"publisher\":\"Coll. Publ., London\",\"year\":\"2010\",\"mrclass\":\"03F30 (03D25 03D30 03F03)\",\"mrnumber\":\"2856914 (2012i:03165)\",\"urlarxiv\":\"http://arxiv.org/abs/0812.3434\",\"abstract\":\"Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.\",\"bibtex\":\"@incollection {MR2856914,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {Priority arguments and epsilon substitution},\\n BOOKTITLE = {Proofs, categories and computations},\\n    SERIES = {Tributes},\\n    VOLUME = {13},\\n     PAGES = {251--266},\\n PUBLISHER = {Coll. Publ., London},\\n      YEAR = {2010},\\n   MRCLASS = {03F30 (03D25 03D30 03F03)},\\n  MRNUMBER = {2856914 (2012i:03165)},\\nurlarxiv={http://arxiv.org/abs/0812.3434},\\nabstract={Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2856914\",\"id\":\"MR2856914\",\"bibbaseid\":\"towsner-priorityargumentsandepsilonsubstitution-2010\",\"role\":\"author\",\"urls\":{\"Arxiv\":\"http://arxiv.org/abs/0812.3434\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":24,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Avigad\"],\"firstnames\":[\"Jeremy\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Functional interpretation and inductive definitions\",\"journal\":\"J. Symbolic Logic\",\"fjournal\":\"Journal of Symbolic Logic\",\"volume\":\"74\",\"year\":\"2009\",\"number\":\"4\",\"pages\":\"1100–1120\",\"issn\":\"0022-4812\",\"coden\":\"JSYLA6\",\"mrclass\":\"03F10 (03F15 03F35)\",\"mrnumber\":\"2583811 (2011c:03137)\",\"mrreviewer\":\"H. Schwichtenberg\",\"doi\":\"10.2178/jsl/1254748682\",\"urljournal\":\"http://dx.doi.org/10.2178/jsl/1254748682\",\"urlarxiv\":\"http://arxiv.org/abs/0802.1938\",\"abstract\":\"Extending G&ouml;odel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.\",\"bibtex\":\"@article {MR2583811,\\n    AUTHOR = {Avigad, Jeremy and Towsner, Henry},\\n     TITLE = {Functional interpretation and inductive definitions},\\n   JOURNAL = {J. Symbolic Logic},\\n  FJOURNAL = {Journal of Symbolic Logic},\\n    VOLUME = {74},\\n      YEAR = {2009},\\n    NUMBER = {4},\\n     PAGES = {1100--1120},\\n      ISSN = {0022-4812},\\n     CODEN = {JSYLA6},\\n   MRCLASS = {03F10 (03F15 03F35)},\\n  MRNUMBER = {2583811 (2011c:03137)},\\nMRREVIEWER = {H. Schwichtenberg},\\n       DOI = {10.2178/jsl/1254748682},\\n       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1254748682},\\nurlarxiv={http://arxiv.org/abs/0802.1938},\\nabstract={Extending G&ouml;odel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.},\\n}\\n\\n\",\"author_short\":[\"Avigad, J.\",\"Towsner, H.\"],\"key\":\"MR2583811\",\"id\":\"MR2583811\",\"bibbaseid\":\"avigad-towsner-functionalinterpretationandinductivedefinitions-2009\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.2178/jsl/1254748682\",\"Arxiv\":\"http://arxiv.org/abs/0802.1938\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":35,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Convergence of diagonal ergodic averages\",\"journal\":\"Ergodic Theory Dynam. Systems\",\"fjournal\":\"Ergodic Theory and Dynamical Systems\",\"volume\":\"29\",\"year\":\"2009\",\"number\":\"4\",\"pages\":\"1309–1326\",\"issn\":\"0143-3857\",\"mrclass\":\"37A30 (28D15)\",\"mrnumber\":\"2529651 (2010f:37005)\",\"mrreviewer\":\"Bryna Kra\",\"doi\":\"10.1017/S0143385708000722\",\"urljournal\":\"http://dx.doi.org/10.1017/S0143385708000722\",\"urlarxiv\":\"http://arxiv.org/abs/0711.1180\",\"abstract\":\"Tao has recently proved that if T<sub>1</sub>,&hellip;,T<sub>l</sub> are commuting, invertible, measure-preserving transformations on a dynamical system then for any L<sub>&infin;</sub> functions f<sub>1</sub>,&hellip;f<sub>l</sub>, the average (1/N)&Sigma;<sub>n=0</sub><sup>N-1</sup>&Pi;<sub>i&leq;l</sub>f<sub>i</sub> o T<sup>in</sup> converges in the L<sub>2</sub> norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence \\\"backwards\\\". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties. \",\"bibtex\":\"@article {MR2529651,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {Convergence of diagonal ergodic averages},\\n   JOURNAL = {Ergodic Theory Dynam. Systems},\\n  FJOURNAL = {Ergodic Theory and Dynamical Systems},\\n    VOLUME = {29},\\n      YEAR = {2009},\\n    NUMBER = {4},\\n     PAGES = {1309--1326},\\n      ISSN = {0143-3857},\\n   MRCLASS = {37A30 (28D15)},\\n  MRNUMBER = {2529651 (2010f:37005)},\\nMRREVIEWER = {Bryna Kra},\\n       DOI = {10.1017/S0143385708000722},\\n       URLJOURNAL= {http://dx.doi.org/10.1017/S0143385708000722},\\nurlarxiv={http://arxiv.org/abs/0711.1180},\\nabstract={Tao has recently proved that if T<sub>1</sub>,&hellip;,T<sub>l</sub> are commuting, invertible, measure-preserving transformations on a dynamical system then for any L<sub>&infin;</sub> functions f<sub>1</sub>,&hellip;f<sub>l</sub>, the average (1/N)&Sigma;<sub>n=0</sub><sup>N-1</sup>&Pi;<sub>i&leq;l</sub>f<sub>i</sub> o T<sup>in</sup> converges in the L<sub>2</sub> norm.  Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence \\\"backwards\\\". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.\\n},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2529651\",\"id\":\"MR2529651\",\"bibbaseid\":\"towsner-convergenceofdiagonalergodicaverages-2009\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1017/S0143385708000722\",\"Arxiv\":\"http://arxiv.org/abs/0711.1180\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":21,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Ordinal analysis by transformations\",\"journal\":\"Ann. Pure Appl. Logic\",\"fjournal\":\"Annals of Pure and Applied Logic\",\"volume\":\"157\",\"year\":\"2009\",\"number\":\"2-3\",\"pages\":\"269–280\",\"issn\":\"0168-0072\",\"coden\":\"APALD7\",\"mrclass\":\"03F05 (03F15 03F35)\",\"mrnumber\":\"2499713 (2010e:03069)\",\"mrreviewer\":\"Thomas Strahm\",\"doi\":\"10.1016/j.apal.2008.09.011\",\"urljournal\":\"http://dx.doi.org/10.1016/j.apal.2008.09.011\",\"urlpdf\":\"http://www.sas.upenn.edu/~htowsner/KPiMu.pdf\",\"abstract\":\"The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz &Omega;<sub>&mu;</sub>-rule, does not apply to systems much stronger than &Pi;<sup>1</sup><sub>1</sub>-comprehension. In this paper, we propose a new extension of the rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with the strength of a recursively inaccessible ordinal.\",\"bibtex\":\"@article {MR2499713,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {Ordinal analysis by transformations},\\n   JOURNAL = {Ann. Pure Appl. Logic},\\n  FJOURNAL = {Annals of Pure and Applied Logic},\\n    VOLUME = {157},\\n      YEAR = {2009},\\n    NUMBER = {2-3},\\n     PAGES = {269--280},\\n      ISSN = {0168-0072},\\n     CODEN = {APALD7},\\n   MRCLASS = {03F05 (03F15 03F35)},\\n  MRNUMBER = {2499713 (2010e:03069)},\\nMRREVIEWER = {Thomas Strahm},\\n       DOI = {10.1016/j.apal.2008.09.011},\\n       URLJOURNAL= {http://dx.doi.org/10.1016/j.apal.2008.09.011},\\nurlpdf={http://www.sas.upenn.edu/~htowsner/KPiMu.pdf},\\nabstract={The technique of using infinitary rules in an ordinal analysis has been one of the\\nmost productive developments in ordinal analysis. Unfortunately, one of the most\\nadvanced variants, the Buchholz &Omega;<sub>&mu;</sub>-rule, does not apply to systems much stronger\\nthan &Pi;<sup>1</sup><sub>1</sub>-comprehension. In this paper, we propose a new extension of the \\n rule using game-theoretic quantifiers. We apply this to a system of inductive definitions\\nwith the strength of a recursively inaccessible ordinal.},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2499713\",\"id\":\"MR2499713\",\"bibbaseid\":\"towsner-ordinalanalysisbytransformations-2009\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1016/j.apal.2008.09.011\",\"Pdf\":\"http://www.sas.upenn.edu/~htowsner/KPiMu.pdf\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":36,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Avigad\"],\"firstnames\":[\"Jeremy\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Gerhardy\"],\"firstnames\":[\"Philipp\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Local stability of ergodic averages\",\"journal\":\"Trans. Amer. Math. Soc.\",\"fjournal\":\"Transactions of the American Mathematical Society\",\"volume\":\"362\",\"year\":\"2010\",\"number\":\"1\",\"pages\":\"261–288\",\"issn\":\"0002-9947\",\"coden\":\"TAMTAM\",\"mrclass\":\"03F60 (28D05)\",\"mrnumber\":\"2550151 (2011e:03082)\",\"mrreviewer\":\"Josef Berger\",\"doi\":\"10.1090/S0002-9947-09-04814-4\",\"urljournal\":\"http://dx.doi.org/10.1090/S0002-9947-09-04814-4\",\"urlarxiv\":\"http://arxiv.org/abs/0706.1512\",\"abstract\":\"The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A<sub>m</sub> f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are \\\"locally stable\\\" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of \\\"proof mining.\\\"\",\"bibtex\":\"@article {MR2550151,\\n    AUTHOR = {Avigad, Jeremy and Gerhardy, Philipp and Towsner, Henry},\\n     TITLE = {Local stability of ergodic averages},\\n   JOURNAL = {Trans. Amer. Math. Soc.},\\n  FJOURNAL = {Transactions of the American Mathematical Society},\\n    VOLUME = {362},\\n      YEAR = {2010},\\n    NUMBER = {1},\\n     PAGES = {261--288},\\n      ISSN = {0002-9947},\\n     CODEN = {TAMTAM},\\n   MRCLASS = {03F60 (28D05)},\\n  MRNUMBER = {2550151 (2011e:03082)},\\nMRREVIEWER = {Josef Berger},\\n       DOI = {10.1090/S0002-9947-09-04814-4},\\n       URLJOURNAL= {http://dx.doi.org/10.1090/S0002-9947-09-04814-4},\\nurlarxiv={http://arxiv.org/abs/0706.1512},\\nabstract={The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A<sub>m</sub> f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are \\\"locally stable\\\" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of \\\"proof mining.\\\"},\\n}\\n\\n\",\"author_short\":[\"Avigad, J.\",\"Gerhardy, P.\",\"Towsner, H.\"],\"key\":\"MR2550151\",\"id\":\"MR2550151\",\"bibbaseid\":\"avigad-gerhardy-towsner-localstabilityofergodicaverages-2010\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1090/S0002-9947-09-04814-4\",\"Arxiv\":\"http://arxiv.org/abs/0706.1512\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":23,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"Epsilon substitution for transfinite induction\",\"journal\":\"Arch. Math. Logic\",\"fjournal\":\"Archive for Mathematical Logic\",\"volume\":\"44\",\"year\":\"2005\",\"number\":\"4\",\"pages\":\"397–412\",\"issn\":\"0933-5846\",\"coden\":\"AMLOEH\",\"mrclass\":\"03F05 (03F07 03F30)\",\"mrnumber\":\"2139689 (2006c:03083)\",\"mrreviewer\":\"Christian Bennet\",\"doi\":\"10.1007/s00153-004-0241-3\",\"urljournal\":\"http://dx.doi.org/10.1007/s00153-004-0241-3\",\"urlpdf\":\"http://www.sas.upenn.edu/~htowsner/esubTI.pdf\",\"abstract\":\"We apply Mints' technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.\",\"bibtex\":\"@article {MR2139689,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {Epsilon substitution for transfinite induction},\\n   JOURNAL = {Arch. Math. Logic},\\n  FJOURNAL = {Archive for Mathematical Logic},\\n    VOLUME = {44},\\n      YEAR = {2005},\\n    NUMBER = {4},\\n     PAGES = {397--412},\\n      ISSN = {0933-5846},\\n     CODEN = {AMLOEH},\\n   MRCLASS = {03F05 (03F07 03F30)},\\n  MRNUMBER = {2139689 (2006c:03083)},\\nMRREVIEWER = {Christian Bennet},\\n       DOI = {10.1007/s00153-004-0241-3},\\n       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0241-3},\\nurlpdf={http://www.sas.upenn.edu/~htowsner/esubTI.pdf},\\nabstract = {We apply Mints' technique for proving the termination of the epsilon\\nsubstitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.},\\n}\\n\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2139689\",\"id\":\"MR2139689\",\"bibbaseid\":\"towsner-epsilonsubstitutionfortransfiniteinduction-2005\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1007/s00153-004-0241-3\",\"Pdf\":\"http://www.sas.upenn.edu/~htowsner/esubTI.pdf\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":42,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Towsner\"],\"firstnames\":[\"Henry\"],\"suffixes\":[]}],\"title\":\"A realizability interpretation for classical analysis\",\"journal\":\"Arch. Math. Logic\",\"fjournal\":\"Archive for Mathematical Logic\",\"volume\":\"43\",\"year\":\"2004\",\"number\":\"7\",\"pages\":\"891–900\",\"issn\":\"0933-5846\",\"coden\":\"AMLOEH\",\"mrclass\":\"03F35 (03F05)\",\"mrnumber\":\"2096140 (2005i:03070)\",\"mrreviewer\":\"Thomas Strahm\",\"doi\":\"10.1007/s00153-004-0233-3\",\"urljournal\":\"http://dx.doi.org/10.1007/s00153-004-0233-3\",\"urlpdf\":\"http://www.sas.upenn.edu/~htowsner/realize.pdf\",\"abstract\":\"We present a realizability interpretation for classical analysis&mdash;an association of a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of &Pi;<sub>2</sub> sentences &forall; x &exist; y A(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).\",\"bibtex\":\"@article {MR2096140,\\n    AUTHOR = {Towsner, Henry},\\n     TITLE = {A realizability interpretation for classical analysis},\\n   JOURNAL = {Arch. Math. Logic},\\n  FJOURNAL = {Archive for Mathematical Logic},\\n    VOLUME = {43},\\n      YEAR = {2004},\\n    NUMBER = {7},\\n     PAGES = {891--900},\\n      ISSN = {0933-5846},\\n     CODEN = {AMLOEH},\\n   MRCLASS = {03F35 (03F05)},\\n  MRNUMBER = {2096140 (2005i:03070)},\\nMRREVIEWER = {Thomas Strahm},\\n       DOI = {10.1007/s00153-004-0233-3},\\n       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0233-3},\\nurlpdf={http://www.sas.upenn.edu/~htowsner/realize.pdf},\\nabstract={We present a realizability interpretation for classical analysis&mdash;an association\\nof a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of &Pi;<sub>2</sub> sentences &forall; x &exist; y A(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).},\\n}\\n\",\"author_short\":[\"Towsner, H.\"],\"key\":\"MR2096140\",\"id\":\"MR2096140\",\"bibbaseid\":\"towsner-arealizabilityinterpretationforclassicalanalysis-2004\",\"role\":\"author\",\"urls\":{\"Journal\":\"http://dx.doi.org/10.1007/s00153-004-0233-3\",\"Pdf\":\"http://www.sas.upenn.edu/~htowsner/realize.pdf\"},\"metadata\":{\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\"downloads\":58,\"html\":\"\"}],\n      metadata: {\"owner\":\"towsner, h\",\"authorlinks\":{\"towsner, h\":\"https://www.sas.upenn.edu/~htowsner/index.html\"}},\n      ownerID: \"W9X6jnrT5WndXxz4f\"\n    };\n  </script>\n\n  <script type=\"text/javascript\">\n  var site$;\n  if (typeof $ != 'undefined') {\n    console.log('existing jquery: storing and then restoring');\n    site$ = $;\n    $ = null;\n  }\n  </script>\n\n  <script src=\"https://ajax.googleapis.com/ajax/libs/jquery/2.2.4/jquery.min.js\">\n  </script>\n  <script type=\"text/javascript\">\n  if (site$) {\n    console.log('existing jquery: avoiding conflict and restoring');\n    bb$ = $.noConflict(true);\n    $ = site$;\n  } else {\n    bb$ = $;\n  }\n  </script>\n\n  <script src=\"//bibbase.org/js/bibbase.min.js\"\n          type=\"text/javascript\">\n  </script>\n\n  <script type=\"text/javascript\"\n          src=\"https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">\n  </script>\n  <script type=\"text/x-mathjax-config\">\n    MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\\\(','\\\\)']]},\n                        skipTags: [\"body\"],\n                        processClass: \"bibbase_paper\"});\n  </script>\n\n  <style>\n  @media print {\n    .dontprint {\n        display: none !important;\n    }\n\n  .bibbase_group {\n    background-color: #E0E0E0;\n    font-style: italic;\n    font-size: larger;\n    text-shadow: 0px 0px 3px #FFFFFF;\n    border-radius: 4px 4px 4px 4px;\n    -moz-border-radius: 4px 4px 4px 4px;\n    -webkit-border-radius: 4px;\n    padding: 5px 40px 5px;\n    margin-top: 10px;\n    margin-bottom: 5px;\n    cursor: pointer;\n  }\n\n  .bibbase_paper {\n    margin-bottom: 5px;\n  }\n\n  }\n  </style>\n\n  \n  <div style=\"float: right; margin-right: 10px; margin-top: 10px; text-align: right; font-size: 12px\">\n    generated by\n    <a href=\"//bibbase.org\"\n       onclick=\"trackOutboundLink('bibbase', 'logo clicked', window.location.href, '//bibbase.org'); return false;\">\n      <img src=\"//bibbase.org/img/logo.svg\"\n           style=\"vertical-align: middle; margin-left: 3px; height: 16px;\"/\n           alt=\"bibbase.org\"\n           title=\"BibBase – The easiest way to maintain your publications page.\"\n        ></a>\n    \n  </div>\n  \n\n  <div id=\"bibbase_header\" class=\"dontprint\" style=\"\">\n\n    <ul class=\"nav nav-pills\" style=\"margin: 0px;\">\n\n      <li class=\"dropdown\" id=\"groupby_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\">\n          Group by\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li class=\"groupby year\"><a onclick=\"groupby('year')\">\n            Year</a>\n        </li>\n        <li class=\"groupby author\"><a onclick=\"groupby('author_short')\">\n            Author</a>\n        </li>\n        <li class=\"groupby type\"><a onclick=\"groupby('type')\">\n            Type</a>\n        </li>\n        <li class=\"groupby keyword\"><a onclick=\"groupby('keyword')\">\n            Keyword</a>\n        </li>\n        <li class=\"groupby downloads\"><a onclick=\"groupby('downloads')\">\n            Downloads</a>\n        </li>\n      </ul>\n      </li>\n\n\n      <li class=\"dropdown\" id=\"menu_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\" alt=\"controls\">\n          <i class=\"fa fa-reorder\" alt=\"controls\"></i>\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li>\n          <a onclick=\"toggleAll()\">\n            <i class=\"fa fa-chevron-down fa-fw\"></i> Expand/Collapse All\n          </a>\n        </li>\n        <li>\n          <a download=\"papers.bib\" href=\"data:text/bibtex;base64,@article{FernndezDuque2023,
  title = {Metric fixed point theory and partial impredicativity},
  volume = {381},
  ISSN = {1471-2962},
  url = {http://dx.doi.org/10.1098/rsta.2022.0012},
  DOI = {10.1098/rsta.2022.0012},
  urlarxiv = {https://arxiv.org/abs/2302.08874},
  number = {2248},
  journal = {Philosophical Transactions of the Royal Society A: Mathematical,  Physical and Engineering Sciences},
  publisher = {The Royal Society},
  author = {Fernández-Duque,  D. and Shafer,  P. and Towsner,  H. and Yokoyama,  K.},
  year = {2023},
  month = apr ,
  abstract = {    We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in 𝖱𝖢𝖠0. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between 𝖠𝖳𝖱0 and Π11-𝖢𝖠0. We also exhibit several weakenings of Caristi's theorem that are equivalent to 𝖶𝖪𝖫0 and to 𝖠𝖢𝖠0. }
}



@misc{2306.12239,
Author = {Henry Towsner},
Title = {From Saturated Embedding Tests to Explicit Algorithms},
Year = {2023},
Eprint = {arXiv:2306.12239},
urlarxiv = {https://arxiv.org/abs/2306.12239},
abstract = {     Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm.
    We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and Günaydin. },
}



@misc{2306.03057,
Author = {Henry Towsner},
Title = {An Aldous--Hoover Theorem for Radon Distributions},
Year = {2023},
Eprint = {arXiv:2306.03057},
urlarxiv = {https://arxiv.org/abs/2306.03057},
abstract = {     We show that the Aldous--Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam \cite{alam2023generalizing} who showed a similar generalization of the di Finetti--Hewitt-Savage Theorem. },
}



@article{CombinatoricsFails,
  doi = {10.1142/s0219061322500234},
  url = {https://doi.org/10.1142/s0219061322500234},
  year = {2022},
  month = dec,
  publisher = {World Scientific Pub Co Pte Ltd},
  author = {Henry Towsner and Rose Weisshaar and Linda Westrick},
  title = {Borel combinatorics fail in HYP},
  journal = {Journal of Mathematical Logic},
  urlarxiv = {https://arxiv.org/abs/2106.13330},
  abstract={     We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author. },
}




@misc{towsner2021removal,
      title={A Removal Lemma for Ordered Hypergraphs}, 
      author={Henry Towsner},
note={draft},
      year={2021},
      eprint={2101.09769},
      archivePrefix={arXiv},
      primaryClass={math.CO},
      urlarxiv={https://arxiv.org/abs/2101.09769},
      abstract={    We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,<) has few induced copies of a small ordered hypergraph (W,H,≺) then there is a small modification G′ so that (V,G′,<) has no induced copies of (W,H,≺). (Note that we do \emph{not} need to modify the ordering <.)
      
    We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of σ-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon. },
}



@misc{chernikov2020hypergraph,
      title={Hypergraph regularity and higher arity VC-dimension}, 
      author={Artem Chernikov and Henry Towsner},
note={draft},
      year={2020},
      eprint={2010.00726},
      archivePrefix={arXiv},
      primaryClass={math.CO},
      urlarxiv={https://arxiv.org/abs/2010.00726},
      abstract={ We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets.
In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels---one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples---and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1.
We also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices. },
}



@article{2018arXiv180804805S,
   author = {{Simmons}, W. and {Towsner}, H.},
    title = "{Explicit polynomial bounds on prime ideals in polynomial rings over fields}",
    month = {August},
    year = {2020},
    urlarxiv={https://arxiv.org/abs/1808.04805},
    journal={Pacific Journal of Mathematics},
    vol = {306},
    issue = {2},
    doi = {10.2140/pjm.2020.306.721},
    pages = {721--754},
    abstract={Suppose I is an ideal of a polynomial ring over a field, I⊆k[x1,…,xn], and whenever fg∈I with degree ≤b, then either f∈I or g∈I. When b is sufficiently large, it follows that I is prime. Schmidt-G\"ottsch proved that "sufficiently large" can be taken to be a polynomial in the degree of generators of I (with the degree of this polynomial depending on n). However Schmidt-G\"ottsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on b, polynomial in the degree of the generators of I. We also give a similar bound for detecting maximal ideals in k[x1,…,xn].}
}



@misc{1807.04452,
author = {{Towsner}, H. and {Yokoyama}, K.},
title="{Erdos-Moser and ISigma_2}",
note={draft},
year=2018,
month=jul,
urlarxiv={https://arxiv.org/abs/1807.04452},
abstract={The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erd\H{o}s-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. 
Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.}}




@misc{EaswarenTowsner,
author = {{Easwaren}, K. and {Towsner}, H.},
title = "{Realism in Mathematics: The Case of the Hyperreals}",
note={submitted},
year=2018,
month= jun,
urlarxiv={https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1},
abstract={A traditional question in the philosophy of mathematics is whether abstract
mathematical objects really exist, just as planets and atoms and giraffes do,
independently of the human mind. Views on this question are often grouped into
“platonist”, “nominalist”, and “constructivist” groups depending on whether
they state that mathematical objects have their own independent existence, or
don’t truly exist at all, or exist but only as constructions of the human mind.
This paper will not directly address this question. We expect that to the extent
that each of these views is defensible, they will not settle the questions that we
are interested in (though there may be some affinities between some possible
answers to these questions). Rather, we will focus on two other questions of
mathematical realism: the question of which mathematical claims can be taken
as meaningful and true within mathematics, and which mathematical ideas can
be taken as applying to the physical world as part of a scientific theory.
We will focus on these questions for the case of the “hyperreals” of nonstandard
analysis, introduced by Abraham Robinson in the 1960’s (which we
will describe in greater detail later). Because of their mathematical properties,
the hyperreals have attracted attacks and defenses in ways that are uncommon
for mathematical entities. Our central claim is that many of the views on both
sides of these debates conflate two distinct philosophical questions. Defenders
of the hyperreals give realist answers to both questions, while attackers give
antirealist answers to both. We defend a realist answer to one question and
an antirealist answer to the other. We think the distinction between these two
questions is important not just for thinking about the hyperreals, but for other
related mathematical entities, and particularly those whose existence depend
on the Axiom of Choice.}
}




@misc{1804.10809,
   author = {{Towsner}, H.},
    title = "{What do ultraproducts remember about the original structures?}",
    note={draft},
     year = 2018,
    month = apr,
urlarxiv={https://arxiv.org/abs/1804.10809},
abstract={We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.}
}




@article{1801.07186,
   author = {{Bernshteyn}, A. and {Delcourt}, M. and {Towsner}, H. and {Tserunyan}, A.
	},
    title = "{A short nonalgorithmic proof of the containers theorem for hypergraphs}",
vol=147,
pages={1739--1749},
     year = 2019,
   urlarxiv = {https://arxiv.org/abs/1801.07186},
   urljournal={https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/},
   journal = {Proc. Amer. Math. Soc},
   doi = {10.1090/proc/14368},
abstract={Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogues of a variety of classical problems from combinatorics and number theory. In this paper, we give the first known proof of this theorem that does not involve an inductive construction (i.e., the so-called scythe algorithm). This proof is less than 4 pages long while being entirely self-contained. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets.
}
}




@misc{1709.05687,
author ={Henry Towsner},
title = {More or Less Uniform Convergence},
year={2017},
month= sep,
note={submitted},
urlarxiv={https://arxiv.org/abs/1709.05687},
abstract={Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals.

We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract alpha-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. 

The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when "proof mining" convergence proofs to obtain quantitative bounds. 

We show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.}
}




@article{1705.10355,
author = {Henry Towsner},
title = {Nonstandard Convergence Gives Bounds on Jumps},
year = {2019},
month = aug,
journal = {New York Journal of Mathematics},
vol = {25},
pages = {651--667},
url = {http://nyjm.albany.edu/j/2019/25-31.html},
urlarxiv={https://arxiv.org/abs/1705.10355},
abstract={If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform "meta-stable" rate of convergence. We show that obtaining a stronger bound---a uniform bound on the number of jumps the sequence makes---is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.}
}



@article{1701.07928,
   author = {I. Goldbring and B. Hart and H. Towsner},
    title = {Explicit sentences distinguishing McDuff's II$\_1$ factors},
    urlarxiv={https://arxiv.org/abs/1701.07928},
    journal={Israel Journal of Mathematics},
 doi = {10.1007/s11856-018-1735-8},
year=2018,
month={Aug},
day=01,
volume=227,
number=1,
pages={365--377},
issn={1565-8511},
urljournal={https://doi.org/10.1007/s11856-018-1735-8},
    abstract={Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable $\Pi_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\"isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.}
}



@article{1609.07509,
author={William Simmons and Henry Towsner},
title={Proof mining and effective bounds in differential polynomial rings},
journal = {Advances in Mathematics},
volume = 343,
pages = {567 - 623},
year = 2019,
issn = {0001-8708},
doi = {https://doi.org/10.1016/j.aim.2018.11.026},
urljournal = {http://www.sciencedirect.com/science/article/pii/S0001870818304900},
urlarxiv={http://arxiv.org/abs/1609.07509},
abstract={Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.}
}



@article{1609.05509,
author = {Henry Towsner},
title = {Constructing Sequences One Step at a Time},
journal = {Journal of Mathematical Logic},
year = {2020},
urlarxiv={http://arxiv.org/abs/1609.05509},
doi={10.1142/S0219061320500178},
urljournal={ https:///dx.doi.org/10.1142/S0219061320500178 },
volume=20,
issue=3,

abstract = {We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle "the product of well-quasi-orders is a well-quasi-order" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.}
}



@misc{hanf_locality,
author = {Steven Lindell and Henry Towsner and Scott Weinstein},
title = {Hanf Locality and Invariant Elementary Definability},
year = {2016},
note = {submitted},
abstract = {We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical rela tions. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call
such expansions presentations of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect
to elementary, neighborhood-bounded presentations. The second is a non-uniform version of the Fagin-
Stockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, local presentations.}
}



@article{1605.04484,
Author = {Harry Crane and Henry Towsner},
Title = {Relative exchangeability with equivalence relations},
Year = {2018},
volume = {57},
issue = {5},
pages={533-556},
journal={Archive for Mathematical Logic},
Eprint = {arXiv:1605.04484},
doi = {10.1007/s00153-017-0591-2},
urlarxiv={http://arxiv.org/abs/1605.04484},
urljournal={http://link.springer.com/article/10.1007/s00153-017-0591-2},
abstract={We describe an Aldous--Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \cite{Ackerman2015} and Crane and Towsner \cite{CraneTowsner2015} and hierarchical exchangeability results due to Austin and Panchenko \cite{AustinPanchenko2014}.
}
}



@misc{1603.05954,
Author = {Harry Crane and Henry Towsner},
Title = {The structure of combinatorial Markov processes},
Year = {2016},
Eprint = {arXiv:1603.05954},
note={submitted},
urlarxiv={http://arxiv.org/abs/1603.05954},
abstract={ Every exchangeable Feller process taking values in a suitably
nice combinatorial state space can be constructed by a system of
iterated random Lipschitz functions. In discrete time, the construction
proceeds by iterated application of independent, identically distributed
functions, while in continuous time the random functions occur as the
atoms of a time homogeneous Poisson point process. We further show
that every exchangeable Feller process projects to a Feller process in an
appropriate limit space, akin to the projection of partition-valued processes
into the ranked-simplex and graph-valued processes into the space
of graph limits. Together, our main theorems establish common structural
features shared by all exchangeable combinatorial Feller processes,
regardless of the dynamics or resident state space, thereby generalizing
behaviors previously observed for exchangeable coalescent and fragmentation
processes as well as other combinatorial stochastic processes. If,
in addition, an exchangeable Feller process evolves on a state space satisfying
the n-disjoint amalgamation property for all n >= 1, then its
jump measure can be decomposed explicitly in the sense of Lévy–Itô–
Khintchine.}
}




@ARTICLE{towsner15:epsilon_cutelim,
author = {{Towsner}, H.},
title="{Epsilon Substitution for $ID_1$ via Cut-Elimination}",
year=2018,
volume = {57},
issue = {5},
pages={497-531},
journal={Archive for Mathematical Logic},
doi = {10.1007/s00153-017-0590-3},
urlarxiv={http://arxiv.org/abs/1509.00390},
urljournal={https://link.springer.com/article/10.1007/s00153-017-0590-3},
abstract={The epsilon-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory ID_1 using a variant of the cut-elimination formalism introduced by Mints.}
}




@ARTICLE{2015arXiv150906733C,
   author = {{Crane}, H. and {Towsner}, H.},
    title = "{Relatively exchangeable structures}",
journal={Journal of Symbolic Logic},
year = 2018,
volume={83},
number={2},
pages={416-442},
urlarxiv={http://arxiv.org/abs/1509.06733},
urljournal={https://doi.org/10.1017/jsl.2017.61},
doi={10.1017/jsl.2017.61},
abstract={We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure &#x1D510;. When &#x1D510; has <i>trivial definable closure</i>, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If &#x1D510; satisfies the stronger properties of <i>ultrahomogeneity</i> and <i>n-disjoint amalgamation property</i> (n-DAP) for every n&geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &#x1D510;.},
}




@ARTICLE{igusa15:computable_ramseys_theorem,
   author = {{Igusa}, G. and {Towsner}, H.},
    title = "{Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets}",
year="2017",
volume="56",
number="1",
    journal = {Archive for Mathematical Logic},
    doi="10.1007/s00153-016-0519-2",
    urljournal={http://link.springer.com/article/10.1007/s00153-016-0519-2},
urlarxiv={http://arxiv.org/abs/1507.03256},
abstract={In [J], Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite &Pi;<sup>0</sup><sup>2</sub> homogeneous set. The proof uses a countable collection of &Pi;<sup>0</sup><sup>2</sub> sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of &Pi;<sup>0</sup><sup>2</sub> sets. We provide a proof of this latter fact.
}
}




@inbook{towsner:abs_cont,
	Author = {Towsner, Henry},
	Chapter = {8},
	Editor = {Patrick C\'egielski, Ali Enayat, and Roman Kossak},
	Publisher = {CSLI},
	Series = {Studies in Weak Arithmetic},
	Title = {Towards an Effective Theory of Absolutely Continuous Measures},
	Volume = {3},
	Year = {2016},
urljournal={https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml},
urlarxiv={http://arxiv.org/abs/1503.04743},
    abstract={We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence L^1 functions. A crucial tool is a one-dimensional version of Szemeredi's regularity lemma for L^1 functions.
}
}



@article{goldbring15:dividing,
year={2015},
issn={0933-5846},
journal={Archive for Mathematical Logic},
volume={54},
number={7-8},
doi={10.1007/s00153-015-0449-4},
title={Dividing and weak quasi-dimensions in arbitrary theories},
url={http://dx.doi.org/10.1007/s00153-015-0449-4},
publisher={Springer Berlin Heidelberg},
keywords={Pseudofinite structure; Quasidimension; Dividing; 03C20; 03C45},
author={Goldbring, Isaac and Towsner, Henry},
pages={915-920},
language={English},
abstract={We show that any countable model of a model complete
theory has an elementary extension with a “pseudofinite-like” quasidimension
that detects dividing.}
}




@article{towsner15:lower_bound_james_space,
   author = {{Towsner}, H.},
    title = "{An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space}",
journal={Houston Journal of Mathematics},
     year = 2018,
    month = aug,
    volume = 44,
    issue = 3,
    pages = {873--885},
    urljournal={https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf},
    urlarxiv={http://arxiv.org/abs/1503.04745},
    abstract={The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.}
}




@unpublished{towsner15:worked_example,
   author = {{Towsner}, H.},
    title = "{A Worked Example of the Functional Interpretation}",
    note={submitted},
     year = 2015,
    month = mar,
   urlarxiv = {http://arxiv.org/abs/1503.05572},
  abstract = {The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.}
}




@article{flood14:separating_principles_below_wkl0,
   author = {{Flood}, S. and {Towsner}, H.},
    title = "{Separating Principles Below WKL0}",
journal={Mathematical Logic Quarterly},
year={2016},
month=dec,
volume = {62},
number = {6},
issn = {1521-3870},
urljournal = {http://dx.doi.org/10.1002/malq.201500001},
doi = {10.1002/malq.201500001},
pages = {507--529},
urlarxiv={http://arxiv.org/abs/1410.4068},
abstract={In this paper, we study Ramsey-type Konig's Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a "one step" construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). 

We illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL from DNR (which has been shown separately by Bienvenu, Patey, and Schafer).
}
}






@article{towsner14:sequence_markov_chains,
	author = {Henry Towsner},
	title = {Limits of sequences of Markov chains},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {20},
	year = {2015},
	keywords = {Markov chain, graph limit, ultraproduct},
	abstract = {We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space.  The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called "random-free" or "product measurable") is simpler than the general case.},
	pages = {no. 77, 1-23},
	issn = {1083-6489},
	doi = {10.1214/EJP.v20-4188},    
        urljournal = {http://ejp.ejpecp.org/article/view/4188},
urlarxiv={http://arxiv.org/abs/1404.3815},
}






@article{towsner13:quasirandom_hypergraphs,
author = {{Towsner}, H.},
title = {Sigma-Algebras for Quasirandom Hypergraphs},
journal = {Random Struct. Alg.},
fjournal = {Random Structures \& Algoritmhs},
volume = {50},
number = {1},
pages = {114--139},
year = {2017},
urljournal={http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract},
doi={10.1002/rsa.20641},
urlarxiv={http://arxiv.org/abs/1312.4882},
abstract = {We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. 
We associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, R\"odl, and Schacht; Conlon, H\`an, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.}
}





@InCollection{lindell13:_infin,
  author = 	 {Steven Lindell and Henry Towsner and Scott Weinstein},
  title = 	 {Infinitary methods in finite model theory},
  booktitle = {Logic Without Borders},
  publisher = {de Gruyter},
  year = 	 2015,
  editor = 	 {Asa Hirvonen and Juha Kontinen and Roman Kossak and Andr\'es Villaveces},
  urlpdf = {http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf},
abstract={
This paper explores applications of the infinitary techniques of classical model theory in
the context of finite model theory, and thus elaborates on themes explored by V&auml;&auml;n&auml;nen
(2003). We are grateful for the opportunity to celebrate Jouko's inclusive vision of logic
in this way.
}
}



@Article{towsner13:maximal_conservatve,
  author = 	 {Towsner, H.},
  title = 	 {On Maximum Conservative Extensions},
  journal = 	 {Computability},
  year = 	 2015,
  volume = 	 4,
  pages = 	 {57--68},
urlarxiv={http://arxiv.org/abs/1302.1488},
abstract={We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma-n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi-2 complete.}
}




@article{doi:10.1142/S0219061313500074,
author = {{Lerman}, M. and {Solomon}, R. and {Towsner}, H.},
title = {Separating principles below Ramsey's Theorem for Pairs},
journal={J. Math. Logic},
fjournal={Journal of Mathematical Logic},
volume = {13},
number = {02},
pages = {1350007},
year = {2013},
doi = {10.1142/S0219061313500074},
URLjournal = {http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074},
urlarxiv={http://arxiv.org/abs/1302.0828},
abstract={In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.},
}





@article {franklin12:_random_non_system,
    AUTHOR = {Franklin, Johanna N. Y. and Towsner, Henry},
     TITLE = {Randomness and non-ergodic systems},
   JOURNAL = {Mosc. Math. J.},
  FJOURNAL = {Moscow Mathematical Journal},
    VOLUME = {14},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {711--744, 827},
      ISSN = {1609-3321},
   MRCLASS = {37A25 (03D32)},
  MRNUMBER = {3292047},
urlarxiv={http://arxiv.org/abs/1206.2682},
urljournal={http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html},
abstract={We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.},
}




@article{towsner12:_analy_approac_spars_hyper,
  author = 	 {Towsner, Henry},
  title = 	 {An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal},
  year = 	 2018,
  journal={Discrete Analysis},
  number={4},
urlarxiv={http://arxiv.org/abs/1204.1884},
urljournal={http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal},
abstract={The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.},
}




@article{doi:10.1142/S0219061314500019,
author = {Towsner, Henry},
title = {Ultrafilters in reverse mathematics},
journal={J. Math. Logic},
fjournal={Journal of Mathematical Logic},
volume = {14},
number = {01},
pages = {1450001},
year = {2014},
doi = {10.1142/S0219061314500019},
urlarxiv={http://arxiv.org/abs/1109.3902},
abstract={We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA<sub>0</sub>, ATR<sub>0</sub>, and &Pi;<sup>1</sup><sub>1</sub>-Comprehension.},
urljournal = {http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019}
}





@article{towsner11:_approx_impred_rever_mathem,
  author = 	 {Towsner, Henry},
  title = 	 {Partial Impredicativity in Reverse Mathematics},
journal={J. Symbolic Logic},
fjournal={Journal of Symbolic Logic},
volume={78},
number={2},
pages={459--488},
urljournal={http://projecteuclid.org/euclid.jsl/1368627060},
  year = 	 2013,
urlarxiv={http://arxiv.org/abs/1106.6063},
abstract={In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a &Pi;<sup>1</sup><sub>2</sub> sentence from the theory &Pi;<sup>1</sup><sub>1</sub>-Comprehension. Using methods based on the functional interpretation, we introduce a family of weakenings of &Pi;<sup>1</sup><sub>1</sub>-Comprehension and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.}
}



@article{goldbring:_approx_logic_measur,
year={2014},
issn={0021-2172},
journal={Israel J. Math.},
fjournal={Israel Journal of Mathematics},
volume={199},
number={2},
doi={10.1007/s11856-013-0054-3},
title={An approximate logic for measures},
urljournal={http://dx.doi.org/10.1007/s11856-013-0054-3},
publisher={The Hebrew University Magnes Press},
author={Goldbring, Isaac and Towsner, Henry},
pages={867-913},
language={English},
urlarxiv={http://arxiv.org/abs/1106.2854},
abstract={We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer&eacute;di Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.},
}




@article {MR2795548,
    AUTHOR = {Towsner, Henry},
     TITLE = {A combinatorial proof of the dense {H}indman's theorem},
   JOURNAL = {Discrete Math.},
  FJOURNAL = {Discrete Mathematics},
    VOLUME = {311},
      YEAR = {2011},
    NUMBER = {14},
     PAGES = {1380--1384},
      ISSN = {0012-365X},
     CODEN = {DSMHA4},
   MRCLASS = {05D10 (03C20)},
  MRNUMBER = {2795548 (2012h:05349)},
MRREVIEWER = {Damir D. Dzhafarov},
       DOI = {10.1016/j.disc.2011.03.006},
       URLJOURNAL= {http://dx.doi.org/10.1016/j.disc.2011.03.006},
urlarxiv={http://arxiv.org/abs/1002.0347},
abstract={The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a "dense" set B<sub>1</sub>, for each b<sub>1</sub>&isin; B<sub>1</sub> a "dense" set B<sub>2</sub>(b<sub>1</sub>) (depending on b<sub>1</sub>), for each b<sub>2</sub>&isin; B<sub>2</sub>(b<sub>1</sub>) a "dense" set B<sub>3</sub>(b<sub>1</sub>, b<sub>2</sub>) (depending on b<sub>1</sub>, b<sub>2</sub>), and so on, such that for any such sequence of b<sub>i</sub>, all finite sums belong to the chosen color. (Here density is often taken to be "piecewise syndetic", but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.}
}




@article {MR2970862,
    AUTHOR = {Beiglbock, Mathias and Towsner, Henry},
     TITLE = {Transfinite approximation of {H}indman's theorem},
   JOURNAL = {Israel J. Math.},
  FJOURNAL = {Israel Journal of Mathematics},
    VOLUME = {191},
      YEAR = {2012},
     PAGES = {41--59},
      ISSN = {0021-2172},
     CODEN = {ISJMAP},
   MRCLASS = {Preliminary Data},
  MRNUMBER = {2970862},
       DOI = {10.1007/s11856-011-0195-1},
       URLJOURNAL = {http://dx.doi.org/10.1007/s11856-011-0195-1},
urlarxiv={http://arxiv.org/abs/1001.1175},
abstract={Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.},
}



@Unpublished{towsner09:_metas_furst_zimmer_tower_ii,
  author = 	 {Towsner, Henry},
  title = 	 {Metastability in the {F}urstenberg-{Z}immer Tower II: Polynomial and Multidimensional {S}zemer\'edi's Theorem},
  note = 	 {draft},
  year = 	 2009,
urlarxiv={http://arxiv.org/abs/0909.5668},
abstract={The Furstenberg-Zimmer structure theorem for &integers;<sub>d,</sub> actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemer\'edi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, &omega;<sup>&omega;<sup>&omega;<sup>&omega;</sup></sup></sup>).
}
}



@Article{towsner12:_simpl_proof_some_diffic_examp_hindm_theor,
  author = 	 {Towsner, Henry},
  title = 	 {A Simple Proof and Some Difficult Examples for {H}indman's Theorem},
journal = {Notre Dame J. Formal Logic},
  fjournal = 	 {Notre Dame Journal of Formal Logic},
  year = 	 2012,
  volume = 	 53,
  number = 	 1,
  pages = 	 {53-65},
urljournal={http://projecteuclid.org/euclid.ndjfl/1336586237},
urlarxiv={http://arxiv.org/abs/0906.3885},
abstract={We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.},
}



@article {MR2791353,
    AUTHOR = {Towsner, Henry},
     TITLE = {Hindman's theorem: an ultrafilter argument in second order
              arithmetic},
   JOURNAL = {J. Symbolic Logic},
  FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {76},
      YEAR = {2011},
    NUMBER = {1},
     PAGES = {353--360},
      ISSN = {0022-4812},
     CODEN = {JSYLA6},
   MRCLASS = {03F35 (54D80)},
  MRNUMBER = {2791353 (2012b:03165)},
MRREVIEWER = {M. Yasuhara},
       DOI = {10.2178/jsl/1294171005},
       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1294171005},
urlarxiv={http://arxiv.org/abs/0906.3882},
abstract={Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.},
}





@Article{towsner12:_corres_princ_gower_unifor_norms,
  author = 	 {Towsner, Henry},
  title = 	 {A Correspondence Principle for the {G}owers Uniformity Norms},
  journal = 	 {Journal of Logic and Analysis},
  year = 	 2012,
  volume = 	 4,
  month = 	 {February},
urljournal={http://logicandanalysis.org/index.php/jla/article/view/139/51},
urlarxiv={http://arxiv.org/abs/0905.0493},
abstract={We give a variation of the Furstenberg Correspondence which preserves the Gowers uniformity norms.},
}



@article {MR2733051,
    AUTHOR = {Avigad, Jeremy and Towsner, Henry},
     TITLE = {Metastability in the {F}urstenberg-{Z}immer tower},
   JOURNAL = {Fund. Math.},
  FJOURNAL = {Fundamenta Mathematicae},
    VOLUME = {210},
      YEAR = {2010},
    NUMBER = {3},
     PAGES = {243--268},
      ISSN = {0016-2736},
   MRCLASS = {37A05 (03E15 03F03 28D05 37A25)},
  MRNUMBER = {2733051 (2012b:37003)},
MRREVIEWER = {Michael H. Schraudner},
       DOI = {10.4064/fm210-3-2},
       URLJOURNAL= {http://dx.doi.org/10.4064/fm210-3-2},
urlarxiv={http://arxiv.org/abs/0902.0356},
abstract={According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer&eacute;di's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the &\omega;<sup>&\omega;<sup>&\omega;</sup></sup>th level.}
}



@incollection {MR2856914,
    AUTHOR = {Towsner, Henry},
     TITLE = {Priority arguments and epsilon substitution},
 BOOKTITLE = {Proofs, categories and computations},
    SERIES = {Tributes},
    VOLUME = {13},
     PAGES = {251--266},
 PUBLISHER = {Coll. Publ., London},
      YEAR = {2010},
   MRCLASS = {03F30 (03D25 03D30 03F03)},
  MRNUMBER = {2856914 (2012i:03165)},
urlarxiv={http://arxiv.org/abs/0812.3434},
abstract={Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.},
}



@article {MR2583811,
    AUTHOR = {Avigad, Jeremy and Towsner, Henry},
     TITLE = {Functional interpretation and inductive definitions},
   JOURNAL = {J. Symbolic Logic},
  FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {74},
      YEAR = {2009},
    NUMBER = {4},
     PAGES = {1100--1120},
      ISSN = {0022-4812},
     CODEN = {JSYLA6},
   MRCLASS = {03F10 (03F15 03F35)},
  MRNUMBER = {2583811 (2011c:03137)},
MRREVIEWER = {H. Schwichtenberg},
       DOI = {10.2178/jsl/1254748682},
       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1254748682},
urlarxiv={http://arxiv.org/abs/0802.1938},
abstract={Extending G&ouml;odel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.},
}



@article {MR2529651,
    AUTHOR = {Towsner, Henry},
     TITLE = {Convergence of diagonal ergodic averages},
   JOURNAL = {Ergodic Theory Dynam. Systems},
  FJOURNAL = {Ergodic Theory and Dynamical Systems},
    VOLUME = {29},
      YEAR = {2009},
    NUMBER = {4},
     PAGES = {1309--1326},
      ISSN = {0143-3857},
   MRCLASS = {37A30 (28D15)},
  MRNUMBER = {2529651 (2010f:37005)},
MRREVIEWER = {Bryna Kra},
       DOI = {10.1017/S0143385708000722},
       URLJOURNAL= {http://dx.doi.org/10.1017/S0143385708000722},
urlarxiv={http://arxiv.org/abs/0711.1180},
abstract={Tao has recently proved that if T<sub>1</sub>,&hellip;,T<sub>l</sub> are commuting, invertible, measure-preserving transformations on a dynamical system then for any L<sub>&infin;</sub> functions f<sub>1</sub>,&hellip;f<sub>l</sub>, the average (1/N)&Sigma;<sub>n=0</sub><sup>N-1</sup>&Pi;<sub>i&leq;l</sub>f<sub>i</sub> o T<sup>in</sup> converges in the L<sub>2</sub> norm.  Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.
},
}



@article {MR2499713,
    AUTHOR = {Towsner, Henry},
     TITLE = {Ordinal analysis by transformations},
   JOURNAL = {Ann. Pure Appl. Logic},
  FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {157},
      YEAR = {2009},
    NUMBER = {2-3},
     PAGES = {269--280},
      ISSN = {0168-0072},
     CODEN = {APALD7},
   MRCLASS = {03F05 (03F15 03F35)},
  MRNUMBER = {2499713 (2010e:03069)},
MRREVIEWER = {Thomas Strahm},
       DOI = {10.1016/j.apal.2008.09.011},
       URLJOURNAL= {http://dx.doi.org/10.1016/j.apal.2008.09.011},
urlpdf={http://www.sas.upenn.edu/~htowsner/KPiMu.pdf},
abstract={The technique of using infinitary rules in an ordinal analysis has been one of the
most productive developments in ordinal analysis. Unfortunately, one of the most
advanced variants, the Buchholz &Omega;<sub>&mu;</sub>-rule, does not apply to systems much stronger
than &Pi;<sup>1</sup><sub>1</sub>-comprehension. In this paper, we propose a new extension of the 
 rule using game-theoretic quantifiers. We apply this to a system of inductive definitions
with the strength of a recursively inaccessible ordinal.},
}



@article {MR2550151,
    AUTHOR = {Avigad, Jeremy and Gerhardy, Philipp and Towsner, Henry},
     TITLE = {Local stability of ergodic averages},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {362},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {261--288},
      ISSN = {0002-9947},
     CODEN = {TAMTAM},
   MRCLASS = {03F60 (28D05)},
  MRNUMBER = {2550151 (2011e:03082)},
MRREVIEWER = {Josef Berger},
       DOI = {10.1090/S0002-9947-09-04814-4},
       URLJOURNAL= {http://dx.doi.org/10.1090/S0002-9947-09-04814-4},
urlarxiv={http://arxiv.org/abs/0706.1512},
abstract={The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A<sub>m</sub> f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."},
}



@article {MR2139689,
    AUTHOR = {Towsner, Henry},
     TITLE = {Epsilon substitution for transfinite induction},
   JOURNAL = {Arch. Math. Logic},
  FJOURNAL = {Archive for Mathematical Logic},
    VOLUME = {44},
      YEAR = {2005},
    NUMBER = {4},
     PAGES = {397--412},
      ISSN = {0933-5846},
     CODEN = {AMLOEH},
   MRCLASS = {03F05 (03F07 03F30)},
  MRNUMBER = {2139689 (2006c:03083)},
MRREVIEWER = {Christian Bennet},
       DOI = {10.1007/s00153-004-0241-3},
       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0241-3},
urlpdf={http://www.sas.upenn.edu/~htowsner/esubTI.pdf},
abstract = {We apply Mints' technique for proving the termination of the epsilon
substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.},
}



@article {MR2096140,
    AUTHOR = {Towsner, Henry},
     TITLE = {A realizability interpretation for classical analysis},
   JOURNAL = {Arch. Math. Logic},
  FJOURNAL = {Archive for Mathematical Logic},
    VOLUME = {43},
      YEAR = {2004},
    NUMBER = {7},
     PAGES = {891--900},
      ISSN = {0933-5846},
     CODEN = {AMLOEH},
   MRCLASS = {03F35 (03F05)},
  MRNUMBER = {2096140 (2005i:03070)},
MRREVIEWER = {Thomas Strahm},
       DOI = {10.1007/s00153-004-0233-3},
       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0233-3},
urlpdf={http://www.sas.upenn.edu/~htowsner/realize.pdf},
abstract={We present a realizability interpretation for classical analysis&mdash;an association
of a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of &Pi;<sub>2</sub> sentences &forall; x &exist; y A(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).},
}
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April 2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1098/rsta.2022.0012\"\n     onclick=\"log_download('fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023', 'http://dx.doi.org/10.1098/rsta.2022.0012', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Metric fixed point theory and partial impredicativity [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://arxiv.org/abs/2302.08874\"\n     onclick=\"log_download('fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023', 'https://arxiv.org/abs/2302.08874', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Metric fixed point theory and partial impredicativity [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1098/rsta.2022.0012\"\n     onclick=\"log_download('fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023', 'http://doi.org/10.1098/rsta.2022.0012', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>13 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('fernndezduque-shafer-towsner-yokoyama-metricfixedpointtheoryandpartialimpredicativity-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{FernndezDuque2023,\n  title = {Metric fixed point theory and partial impredicativity},\n  volume = {381},\n  ISSN = {1471-2962},\n  url = {http://dx.doi.org/10.1098/rsta.2022.0012},\n  DOI = {10.1098/rsta.2022.0012},\n  urlarxiv = {https://arxiv.org/abs/2302.08874},\n  number = {2248},\n  journal = {Philosophical Transactions of the Royal Society A: Mathematical,  Physical and Engineering Sciences},\n  publisher = {The Royal Society},\n  author = {Fernández-Duque,  D. and Shafer,  P. and Towsner,  H. and Yokoyama,  K.},\n  year = {2023},\n  month = apr ,\n  abstract = {    We show that the Priess-Crampe &amp; Ribenboim fixed point theorem is provable in 𝖱𝖢𝖠0. Furthermore, we show that Caristi&#x27;s fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between 𝖠𝖳𝖱0 and Π11-𝖢𝖠0. We also exhibit several weakenings of Caristi&#x27;s theorem that are equivalent to 𝖶𝖪𝖫0 and to 𝖠𝖢𝖠0. }\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in 𝖱𝖢𝖠0. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between 𝖠𝖳𝖱0 and Π11-𝖢𝖠0. We also exhibit several weakenings of Caristi's theorem that are equivalent to 𝖶𝖪𝖫0 and to 𝖠𝖢𝖠0. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2306.12239\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023', 'https://arxiv.org/abs/2306.12239', event);\">\n    From Saturated Embedding Tests to Explicit Algorithms.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2306.12239\"\n     onclick=\"log_download('towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023', 'https://arxiv.org/abs/2306.12239', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"From Saturated Embedding Tests to Explicit Algorithms [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>12 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-fromsaturatedembeddingteststoexplicitalgorithms-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{2306.12239,\nAuthor = {Henry Towsner},\nTitle = {From Saturated Embedding Tests to Explicit Algorithms},\nYear = {2023},\nEprint = {arXiv:2306.12239},\nurlarxiv = {https://arxiv.org/abs/2306.12239},\nabstract = {     Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm.\n    We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and Günaydin. },\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form – usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm. We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and Günaydin. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-analdoushoovertheoremforradondistributions-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2306.03057\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-analdoushoovertheoremforradondistributions-2023', 'https://arxiv.org/abs/2306.03057', event);\">\n    An Aldous–Hoover Theorem for Radon Distributions.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2306.03057\"\n     onclick=\"log_download('towsner-analdoushoovertheoremforradondistributions-2023', 'https://arxiv.org/abs/2306.03057', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An Aldous–Hoover Theorem for Radon Distributions [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-analdoushoovertheoremforradondistributions-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-analdoushoovertheoremforradondistributions-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{2306.03057,\nAuthor = {Henry Towsner},\nTitle = {An Aldous--Hoover Theorem for Radon Distributions},\nYear = {2023},\nEprint = {arXiv:2306.03057},\nurlarxiv = {https://arxiv.org/abs/2306.03057},\nabstract = {     We show that the Aldous--Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam \\cite{alam2023generalizing} who showed a similar generalization of the di Finetti--Hewitt-Savage Theorem. },\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We show that the Aldous–Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam i̧tealam2023generalizing who showed a similar generalization of the di Finetti–Hewitt-Savage Theorem. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2022\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2022')\"\n      onkeypress=\"toggleGroup('group_2022')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2022</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://doi.org/10.1142/s0219061322500234\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022', 'https://doi.org/10.1142/s0219061322500234', event);\">\n    Borel combinatorics fail in HYP.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>; Weisshaar, R.;  and Westrick, L.\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Mathematical Logic</i>. December 2022.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://doi.org/10.1142/s0219061322500234\"\n     onclick=\"log_download('towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022', 'https://doi.org/10.1142/s0219061322500234', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Borel combinatorics fail in HYP [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://arxiv.org/abs/2106.13330\"\n     onclick=\"log_download('towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022', 'https://arxiv.org/abs/2106.13330', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Borel combinatorics fail in HYP [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1142/s0219061322500234\"\n     onclick=\"log_download('towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022', 'http://doi.org/10.1142/s0219061322500234', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-weisshaar-westrick-borelcombinatoricsfailinhyp-2022')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{CombinatoricsFails,\n  doi = {10.1142/s0219061322500234},\n  url = {https://doi.org/10.1142/s0219061322500234},\n  year = {2022},\n  month = dec,\n  publisher = {World Scientific Pub Co Pte Ltd},\n  author = {Henry Towsner and Rose Weisshaar and Linda Westrick},\n  title = {Borel combinatorics fail in HYP},\n  journal = {Journal of Mathematical Logic},\n  urlarxiv = {https://arxiv.org/abs/2106.13330},\n  abstract={     We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon &amp; the third author. },\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We characterize the completely determined Borel subsets of HYP as exactly the omega_1^ck subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2021\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2021')\"\n      onkeypress=\"toggleGroup('group_2021')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2021</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-aremovallemmafororderedhypergraphs-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2101.09769\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-aremovallemmafororderedhypergraphs-2021', 'https://arxiv.org/abs/2101.09769', event);\">\n    A Removal Lemma for Ordered Hypergraphs.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2021.\n  <span class=\"note\">draft</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2101.09769\"\n     onclick=\"log_download('towsner-aremovallemmafororderedhypergraphs-2021', 'https://arxiv.org/abs/2101.09769', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Removal Lemma for Ordered Hypergraphs [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-aremovallemmafororderedhypergraphs-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>60 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-aremovallemmafororderedhypergraphs-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{towsner2021removal,\n      title={A Removal Lemma for Ordered Hypergraphs}, \n      author={Henry Towsner},\nnote={draft},\n      year={2021},\n      eprint={2101.09769},\n      archivePrefix={arXiv},\n      primaryClass={math.CO},\n      urlarxiv={https://arxiv.org/abs/2101.09769},\n      abstract={    We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer&#x27;s removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,&lt;) has few induced copies of a small ordered hypergraph (W,H,≺) then there is a small modification G′ so that (V,G′,&lt;) has no induced copies of (W,H,≺). (Note that we do \\emph{not} need to modify the ordering &lt;.)\n      \n    We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of σ-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a &#x60;&#x60;very structured&#x27;&#x27; set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon. },\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon–Ben-Eliezer–Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph (V,G,<) has few induced copies of a small ordered hypergraph (W,H,≺) then there is a small modification G′ so that (V,G′,<) has no induced copies of (W,H,≺). (Note that we do \\emphnot need to modify the ordering <.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of σ-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2020\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2020')\"\n      onkeypress=\"toggleGroup('group_2020')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2020</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2010.00726\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020', 'https://arxiv.org/abs/2010.00726', event);\">\n    Hypergraph regularity and higher arity VC-dimension.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Chernikov, A.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2020.\n  <span class=\"note\">draft</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2010.00726\"\n     onclick=\"log_download('chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020', 'https://arxiv.org/abs/2010.00726', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Hypergraph regularity and higher arity VC-dimension [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>21 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('chernikov-towsner-hypergraphregularityandhigherarityvcdimension-2020')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{chernikov2020hypergraph,\n      title={Hypergraph regularity and higher arity VC-dimension}, \n      author={Artem Chernikov and Henry Towsner},\nnote={draft},\n      year={2020},\n      eprint={2010.00726},\n      archivePrefix={arXiv},\n      primaryClass={math.CO},\n      urlarxiv={https://arxiv.org/abs/2010.00726},\n      abstract={ We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets.\nIn the language of hypergraph regularity, this shows that when H is a k&#x27;-uniform hypergraph with small VC_k-dimension for some k&lt;k&#x27;, the decomposition of H given by hypergraph regularity only needs the first k levels---one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples---and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1.\nWe also show a suitable converse: k&#x27;-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices. },\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels—one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples—and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1. We also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1808.04805\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020', 'https://arxiv.org/abs/1808.04805', event);\">\n    Explicit polynomial bounds on prime ideals in polynomial rings over fields.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Simmons, W.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Pacific Journal of Mathematics</i>,721–754. August 2020.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1808.04805\"\n     onclick=\"log_download('simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020', 'https://arxiv.org/abs/1808.04805', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Explicit polynomial bounds on prime ideals in polynomial rings over fields [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.2140/pjm.2020.306.721\"\n     onclick=\"log_download('simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020', 'http://doi.org/10.2140/pjm.2020.306.721', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>16 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('simmons-towsner-explicitpolynomialboundsonprimeidealsinpolynomialringsoverfields-2020')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{2018arXiv180804805S,\n   author = {{Simmons}, W. and {Towsner}, H.},\n    title = &quot;{Explicit polynomial bounds on prime ideals in polynomial rings over fields}&quot;,\n    month = {August},\n    year = {2020},\n    urlarxiv={https://arxiv.org/abs/1808.04805},\n    journal={Pacific Journal of Mathematics},\n    vol = {306},\n    issue = {2},\n    doi = {10.2140/pjm.2020.306.721},\n    pages = {721--754},\n    abstract={Suppose I is an ideal of a polynomial ring over a field, I⊆k[x1,…,xn], and whenever fg∈I with degree ≤b, then either f∈I or g∈I. When b is sufficiently large, it follows that I is prime. Schmidt-G\\&quot;ottsch proved that &quot;sufficiently large&quot; can be taken to be a polynomial in the degree of generators of I (with the degree of this polynomial depending on n). However Schmidt-G\\&quot;ottsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on b, polynomial in the degree of the generators of I. We also give a similar bound for detecting maximal ideals in k[x1,…,xn].}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Suppose I is an ideal of a polynomial ring over a field, I⊆k[x1,…,xn], and whenever fg∈I with degree ≤b, then either f∈I or g∈I. When b is sufficiently large, it follows that I is prime. Schmidt-Göttsch proved that \"sufficiently large\" can be taken to be a polynomial in the degree of generators of I (with the degree of this polynomial depending on n). However Schmidt-Göttsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on b, polynomial in the degree of the generators of I. We also give a similar bound for detecting maximal ideals in k[x1,…,xn].\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-constructingsequencesonestepatatime-2020\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1609.05509\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-constructingsequencesonestepatatime-2020', 'http://arxiv.org/abs/1609.05509', event);\">\n    Constructing Sequences One Step at a Time.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Mathematical Logic</i>, 20.  2020.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1609.05509\"\n     onclick=\"log_download('towsner-constructingsequencesonestepatatime-2020', 'http://arxiv.org/abs/1609.05509', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Constructing Sequences One Step at a Time [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https:///dx.doi.org/10.1142/S0219061320500178 \"\n     onclick=\"log_download('towsner-constructingsequencesonestepatatime-2020', 'https:///dx.doi.org/10.1142/S0219061320500178 ', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Constructing Sequences One Step at a Time [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1142/S0219061320500178\"\n     onclick=\"log_download('towsner-constructingsequencesonestepatatime-2020', 'http://doi.org/10.1142/S0219061320500178', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-constructingsequencesonestepatatime-2020\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>31 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-constructingsequencesonestepatatime-2020')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1609.05509,\nauthor = {Henry Towsner},\ntitle = {Constructing Sequences One Step at a Time},\njournal = {Journal of Mathematical Logic},\nyear = {2020},\nurlarxiv={http://arxiv.org/abs/1609.05509},\ndoi={10.1142/S0219061320500178},\nurljournal={ https:///dx.doi.org/10.1142/S0219061320500178 },\nvolume=20,\nissue=3,\n\nabstract = {We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig&#x27;s Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle &quot;the product of well-quasi-orders is a well-quasi-order&quot; is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle \"the product of well-quasi-orders is a well-quasi-order\" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2019\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2019')\"\n      onkeypress=\"toggleGroup('group_2019')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2019</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1801.07186\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019', 'https://arxiv.org/abs/1801.07186', event);\">\n    A short nonalgorithmic proof of the containers theorem for hypergraphs.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Bernshteyn, A.; Delcourt, M.; <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>;  and Tserunyan, A.\n</span>\n\n  \n\n</span>\n\n  <i>Proc. Amer. Math. Soc</i>,1739–1749.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1801.07186\"\n     onclick=\"log_download('bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019', 'https://arxiv.org/abs/1801.07186', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A short nonalgorithmic proof of the containers theorem for hypergraphs [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/\"\n     onclick=\"log_download('bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019', 'https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A short nonalgorithmic proof of the containers theorem for hypergraphs [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1090/proc/14368\"\n     onclick=\"log_download('bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019', 'http://doi.org/10.1090/proc/14368', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>22 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('bernshteyn-delcourt-towsner-tserunyan-ashortnonalgorithmicproofofthecontainerstheoremforhypergraphs-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1801.07186,\n   author = {{Bernshteyn}, A. and {Delcourt}, M. and {Towsner}, H. and {Tserunyan}, A.\n\t},\n    title = &quot;{A short nonalgorithmic proof of the containers theorem for hypergraphs}&quot;,\nvol=147,\npages={1739--1749},\n     year = 2019,\n   urlarxiv = {https://arxiv.org/abs/1801.07186},\n   urljournal={https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2019-14368-9/},\n   journal = {Proc. Amer. Math. Soc},\n   doi = {10.1090/proc/14368},\nabstract={Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogues of a variety of classical problems from combinatorics and number theory. In this paper, we give the first known proof of this theorem that does not involve an inductive construction (i.e., the so-called scythe algorithm). This proof is less than 4 pages long while being entirely self-contained. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets.\n}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogues of a variety of classical problems from combinatorics and number theory. In this paper, we give the first known proof of this theorem that does not involve an inductive construction (i.e., the so-called scythe algorithm). This proof is less than 4 pages long while being entirely self-contained. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-nonstandardconvergencegivesboundsonjumps-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://nyjm.albany.edu/j/2019/25-31.html\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-nonstandardconvergencegivesboundsonjumps-2019', 'http://nyjm.albany.edu/j/2019/25-31.html', event);\">\n    Nonstandard Convergence Gives Bounds on Jumps.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>New York Journal of Mathematics</i>,651–667. August 2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://nyjm.albany.edu/j/2019/25-31.html\"\n     onclick=\"log_download('towsner-nonstandardconvergencegivesboundsonjumps-2019', 'http://nyjm.albany.edu/j/2019/25-31.html', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Nonstandard Convergence Gives Bounds on Jumps [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://arxiv.org/abs/1705.10355\"\n     onclick=\"log_download('towsner-nonstandardconvergencegivesboundsonjumps-2019', 'https://arxiv.org/abs/1705.10355', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Nonstandard Convergence Gives Bounds on Jumps [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-nonstandardconvergencegivesboundsonjumps-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>9 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-nonstandardconvergencegivesboundsonjumps-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1705.10355,\nauthor = {Henry Towsner},\ntitle = {Nonstandard Convergence Gives Bounds on Jumps},\nyear = {2019},\nmonth = aug,\njournal = {New York Journal of Mathematics},\nvol = {25},\npages = {651--667},\nurl = {http://nyjm.albany.edu/j/2019/25-31.html},\nurlarxiv={https://arxiv.org/abs/1705.10355},\nabstract={If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform &quot;meta-stable&quot; rate of convergence. We show that obtaining a stronger bound---a uniform bound on the number of jumps the sequence makes---is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform \"meta-stable\" rate of convergence. We show that obtaining a stronger bound—a uniform bound on the number of jumps the sequence makes—is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://www.sciencedirect.com/science/article/pii/S0001870818304900\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019', 'http://www.sciencedirect.com/science/article/pii/S0001870818304900', event);\">\n    Proof mining and effective bounds in differential polynomial rings.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Simmons, W.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Advances in Mathematics</i>, 343: 567 - 623.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://www.sciencedirect.com/science/article/pii/S0001870818304900\"\n     onclick=\"log_download('simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019', 'http://www.sciencedirect.com/science/article/pii/S0001870818304900', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Proof mining and effective bounds in differential polynomial rings [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1609.07509\"\n     onclick=\"log_download('simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019', 'http://arxiv.org/abs/1609.07509', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Proof mining and effective bounds in differential polynomial rings [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/https://doi.org/10.1016/j.aim.2018.11.026\"\n     onclick=\"log_download('simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019', 'http://doi.org/https://doi.org/10.1016/j.aim.2018.11.026', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>34 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('simmons-towsner-proofminingandeffectiveboundsindifferentialpolynomialrings-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1609.07509,\nauthor={William Simmons and Henry Towsner},\ntitle={Proof mining and effective bounds in differential polynomial rings},\njournal = {Advances in Mathematics},\nvolume = 343,\npages = {567 - 623},\nyear = 2019,\nissn = {0001-8708},\ndoi = {https://doi.org/10.1016/j.aim.2018.11.026},\nurljournal = {http://www.sciencedirect.com/science/article/pii/S0001870818304900},\nurlarxiv={http://arxiv.org/abs/1609.07509},\nabstract={Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2018\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2018')\"\n      onkeypress=\"toggleGroup('group_2018')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2018</span>\n      <span class=\"bibbase_group_count\">\n        \n        (9)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-yokoyama-erdosmoserandisigma2-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1807.04452\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-yokoyama-erdosmoserandisigma2-2018', 'https://arxiv.org/abs/1807.04452', event);\">\n    Erdos-Moser and ISigma_2.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>;  and Yokoyama, K.\n</span>\n\n  \n\n</span>\n\n  July 2018.\n  <span class=\"note\">draft</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1807.04452\"\n     onclick=\"log_download('towsner-yokoyama-erdosmoserandisigma2-2018', 'https://arxiv.org/abs/1807.04452', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Erdos-Moser and ISigma_2 [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-yokoyama-erdosmoserandisigma2-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>14 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-yokoyama-erdosmoserandisigma2-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{1807.04452,\nauthor = {{Towsner}, H. and {Yokoyama}, K.},\ntitle=&quot;{Erdos-Moser and ISigma_2}&quot;,\nnote={draft},\nyear=2018,\nmonth=jul,\nurlarxiv={https://arxiv.org/abs/1807.04452},\nabstract={The first-order part of the Ramsey&#x27;s Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey&#x27;s Theorem for pairs into the weaker principles, EM (the Erd\\H{o}s-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. \nSpecifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.}}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the Erdős-Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond ISigma02 is entirely due to the arbitrary color analog of ADS. Specifically, we show that ADS for an arbitrary number of colors implies BSigma03 while EM for an arbitrary number of colors is Pi11-conservative over ISigma02 and it does not imply ISigma02.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018', 'https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1', event);\">\n    Realism in Mathematics: The Case of the Hyperreals.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Easwaren, K.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  June 2018.\n  <span class=\"note\">submitted</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1\"\n     onclick=\"log_download('easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018', 'https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Realism in Mathematics: The Case of the Hyperreals [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>26 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('easwaren-towsner-realisminmathematicsthecaseofthehyperreals-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{EaswarenTowsner,\nauthor = {{Easwaren}, K. and {Towsner}, H.},\ntitle = &quot;{Realism in Mathematics: The Case of the Hyperreals}&quot;,\nnote={submitted},\nyear=2018,\nmonth= jun,\nurlarxiv={https://www.dropbox.com/s/vmgevvgmy9bhl2c/Hyperreals.pdf?raw=1},\nabstract={A traditional question in the philosophy of mathematics is whether abstract\nmathematical objects really exist, just as planets and atoms and giraffes do,\nindependently of the human mind. Views on this question are often grouped into\n“platonist”, “nominalist”, and “constructivist” groups depending on whether\nthey state that mathematical objects have their own independent existence, or\ndon’t truly exist at all, or exist but only as constructions of the human mind.\nThis paper will not directly address this question. We expect that to the extent\nthat each of these views is defensible, they will not settle the questions that we\nare interested in (though there may be some affinities between some possible\nanswers to these questions). Rather, we will focus on two other questions of\nmathematical realism: the question of which mathematical claims can be taken\nas meaningful and true within mathematics, and which mathematical ideas can\nbe taken as applying to the physical world as part of a scientific theory.\nWe will focus on these questions for the case of the “hyperreals” of nonstandard\nanalysis, introduced by Abraham Robinson in the 1960’s (which we\nwill describe in greater detail later). Because of their mathematical properties,\nthe hyperreals have attracted attacks and defenses in ways that are uncommon\nfor mathematical entities. Our central claim is that many of the views on both\nsides of these debates conflate two distinct philosophical questions. Defenders\nof the hyperreals give realist answers to both questions, while attackers give\nantirealist answers to both. We defend a realist answer to one question and\nan antirealist answer to the other. We think the distinction between these two\nquestions is important not just for thinking about the hyperreals, but for other\nrelated mathematical entities, and particularly those whose existence depend\non the Axiom of Choice.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  A traditional question in the philosophy of mathematics is whether abstract mathematical objects really exist, just as planets and atoms and giraffes do, independently of the human mind. Views on this question are often grouped into “platonist”, “nominalist”, and “constructivist” groups depending on whether they state that mathematical objects have their own independent existence, or don’t truly exist at all, or exist but only as constructions of the human mind. This paper will not directly address this question. We expect that to the extent that each of these views is defensible, they will not settle the questions that we are interested in (though there may be some affinities between some possible answers to these questions). Rather, we will focus on two other questions of mathematical realism: the question of which mathematical claims can be taken as meaningful and true within mathematics, and which mathematical ideas can be taken as applying to the physical world as part of a scientific theory. We will focus on these questions for the case of the “hyperreals” of nonstandard analysis, introduced by Abraham Robinson in the 1960’s (which we will describe in greater detail later). Because of their mathematical properties, the hyperreals have attracted attacks and defenses in ways that are uncommon for mathematical entities. Our central claim is that many of the views on both sides of these debates conflate two distinct philosophical questions. Defenders of the hyperreals give realist answers to both questions, while attackers give antirealist answers to both. We defend a realist answer to one question and an antirealist answer to the other. We think the distinction between these two questions is important not just for thinking about the hyperreals, but for other related mathematical entities, and particularly those whose existence depend on the Axiom of Choice.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1804.10809\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018', 'https://arxiv.org/abs/1804.10809', event);\">\n    What do ultraproducts remember about the original structures?.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  April 2018.\n  <span class=\"note\">draft</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1804.10809\"\n     onclick=\"log_download('towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018', 'https://arxiv.org/abs/1804.10809', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"What do ultraproducts remember about the original structures? [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>36 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-whatdoultraproductsrememberabouttheoriginalstructures-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{1804.10809,\n   author = {{Towsner}, H.},\n    title = &quot;{What do ultraproducts remember about the original structures?}&quot;,\n    note={draft},\n     year = 2018,\n    month = apr,\nurlarxiv={https://arxiv.org/abs/1804.10809},\nabstract={We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1701.07928\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018', 'https://arxiv.org/abs/1701.07928', event);\">\n    Explicit sentences distinguishing McDuff's II$_1$ factors.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Goldbring, I.; Hart, B.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Israel Journal of Mathematics</i>, 227(1): 365–377. Aug 2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1701.07928\"\n     onclick=\"log_download('goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018', 'https://arxiv.org/abs/1701.07928', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Explicit sentences distinguishing McDuff&#x27;s II$_1$ factors [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://doi.org/10.1007/s11856-018-1735-8\"\n     onclick=\"log_download('goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018', 'https://doi.org/10.1007/s11856-018-1735-8', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Explicit sentences distinguishing McDuff&#x27;s II$_1$ factors [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s11856-018-1735-8\"\n     onclick=\"log_download('goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018', 'http://doi.org/10.1007/s11856-018-1735-8', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('goldbring-hart-towsner-explicitsentencesdistinguishingmcduffsii1factors-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1701.07928,\n   author = {I. Goldbring and B. Hart and H. Towsner},\n    title = {Explicit sentences distinguishing McDuff&#x27;s II$\\_1$ factors},\n    urlarxiv={https://arxiv.org/abs/1701.07928},\n    journal={Israel Journal of Mathematics},\n doi = {10.1007/s11856-018-1735-8},\nyear=2018,\nmonth={Aug},\nday=01,\nvolume=227,\nnumber=1,\npages={365--377},\nissn={1565-8511},\nurljournal={https://doi.org/10.1007/s11856-018-1735-8},\n    abstract={Recently, Boutonnet, Chifan, and Ioana proved that McDuff&#x27;s examples of continuum many pairwise non-isomorphic separable $\\Pi_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fra\\&quot;isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Recently, Boutonnet, Chifan, and Ioana proved that McDuff's examples of continuum many pairwise non-isomorphic separable $Π_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fraïsse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"crane-towsner-relativeexchangeabilitywithequivalencerelations-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1605.04484\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'crane-towsner-relativeexchangeabilitywithequivalencerelations-2018', 'http://arxiv.org/abs/1605.04484', event);\">\n    Relative exchangeability with equivalence relations.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Crane, H.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Archive for Mathematical Logic</i>, 57: 533-556.  2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1605.04484\"\n     onclick=\"log_download('crane-towsner-relativeexchangeabilitywithequivalencerelations-2018', 'http://arxiv.org/abs/1605.04484', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Relative exchangeability with equivalence relations [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"http://link.springer.com/article/10.1007/s00153-017-0591-2\"\n     onclick=\"log_download('crane-towsner-relativeexchangeabilitywithequivalencerelations-2018', 'http://link.springer.com/article/10.1007/s00153-017-0591-2', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Relative exchangeability with equivalence relations [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-017-0591-2\"\n     onclick=\"log_download('crane-towsner-relativeexchangeabilitywithequivalencerelations-2018', 'http://doi.org/10.1007/s00153-017-0591-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/crane-towsner-relativeexchangeabilitywithequivalencerelations-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>11 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('crane-towsner-relativeexchangeabilitywithequivalencerelations-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1605.04484,\nAuthor = {Harry Crane and Henry Towsner},\nTitle = {Relative exchangeability with equivalence relations},\nYear = {2018},\nvolume = {57},\nissue = {5},\npages={533-556},\njournal={Archive for Mathematical Logic},\nEprint = {arXiv:1605.04484},\ndoi = {10.1007/s00153-017-0591-2},\nurlarxiv={http://arxiv.org/abs/1605.04484},\nurljournal={http://link.springer.com/article/10.1007/s00153-017-0591-2},\nabstract={We describe an Aldous--Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \\cite{Ackerman2015} and Crane and Towsner \\cite{CraneTowsner2015} and hierarchical exchangeability results due to Austin and Panchenko \\cite{AustinPanchenko2014}.\n}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman i̧teAckerman2015 and Crane and Towsner i̧teCraneTowsner2015 and hierarchical exchangeability results due to Austin and Panchenko i̧teAustinPanchenko2014. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-epsilonsubstitutionforid1viacutelimination-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1509.00390\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-epsilonsubstitutionforid1viacutelimination-2018', 'http://arxiv.org/abs/1509.00390', event);\">\n    Epsilon Substitution for $ID_1$ via Cut-Elimination.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Archive for Mathematical Logic</i>, 57: 497-531.  2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1509.00390\"\n     onclick=\"log_download('towsner-epsilonsubstitutionforid1viacutelimination-2018', 'http://arxiv.org/abs/1509.00390', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Epsilon Substitution for $ID_1$ via Cut-Elimination [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://link.springer.com/article/10.1007/s00153-017-0590-3\"\n     onclick=\"log_download('towsner-epsilonsubstitutionforid1viacutelimination-2018', 'https://link.springer.com/article/10.1007/s00153-017-0590-3', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Epsilon Substitution for $ID_1$ via Cut-Elimination [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-017-0590-3\"\n     onclick=\"log_download('towsner-epsilonsubstitutionforid1viacutelimination-2018', 'http://doi.org/10.1007/s00153-017-0590-3', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-epsilonsubstitutionforid1viacutelimination-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>9 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-epsilonsubstitutionforid1viacutelimination-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@ARTICLE{towsner15:epsilon_cutelim,\nauthor = {{Towsner}, H.},\ntitle=&quot;{Epsilon Substitution for $ID_1$ via Cut-Elimination}&quot;,\nyear=2018,\nvolume = {57},\nissue = {5},\npages={497-531},\njournal={Archive for Mathematical Logic},\ndoi = {10.1007/s00153-017-0590-3},\nurlarxiv={http://arxiv.org/abs/1509.00390},\nurljournal={https://link.springer.com/article/10.1007/s00153-017-0590-3},\nabstract={The epsilon-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory ID_1 using a variant of the cut-elimination formalism introduced by Mints.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The epsilon-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory ID_1 using a variant of the cut-elimination formalism introduced by Mints.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"crane-towsner-relativelyexchangeablestructures-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1509.06733\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'crane-towsner-relativelyexchangeablestructures-2018', 'http://arxiv.org/abs/1509.06733', event);\">\n    Relatively exchangeable structures.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Crane, H.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Symbolic Logic</i>, 83(2): 416-442.  2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1509.06733\"\n     onclick=\"log_download('crane-towsner-relativelyexchangeablestructures-2018', 'http://arxiv.org/abs/1509.06733', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Relatively exchangeable structures [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://doi.org/10.1017/jsl.2017.61\"\n     onclick=\"log_download('crane-towsner-relativelyexchangeablestructures-2018', 'https://doi.org/10.1017/jsl.2017.61', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Relatively exchangeable structures [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1017/jsl.2017.61\"\n     onclick=\"log_download('crane-towsner-relativelyexchangeablestructures-2018', 'http://doi.org/10.1017/jsl.2017.61', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/crane-towsner-relativelyexchangeablestructures-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>10 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('crane-towsner-relativelyexchangeablestructures-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@ARTICLE{2015arXiv150906733C,\n   author = {{Crane}, H. and {Towsner}, H.},\n    title = &quot;{Relatively exchangeable structures}&quot;,\njournal={Journal of Symbolic Logic},\nyear = 2018,\nvolume={83},\nnumber={2},\npages={416-442},\nurlarxiv={http://arxiv.org/abs/1509.06733},\nurljournal={https://doi.org/10.1017/jsl.2017.61},\ndoi={10.1017/jsl.2017.61},\nabstract={We study random relational structures that are \\emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure &amp;#x1D510;. When &amp;#x1D510; has &lt;i&gt;trivial definable closure&lt;/i&gt;, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If &amp;#x1D510; satisfies the stronger properties of &lt;i&gt;ultrahomogeneity&lt;/i&gt; and &lt;i&gt;n-disjoint amalgamation property&lt;/i&gt; (n-DAP) for every n&amp;geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &amp;#x1D510;.},\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We study random relational structures that are \\emphrelatively exchangeable—that is, whose distributions are invariant under the automorphisms of a reference structure &#x1D510;. When &#x1D510; has <i>trivial definable closure</i>, every relatively exchangeable structure satisfies a general Aldous–Hoover-type representation. If &#x1D510; satisfies the stronger properties of <i>ultrahomogeneity</i> and <i>n-disjoint amalgamation property</i> (n-DAP) for every n&geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &#x1D510;.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018', 'https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf', event);\">\n    An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Houston Journal of Mathematics</i>, 44: 873–885. August 2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf\"\n     onclick=\"log_download('towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018', 'https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space [pdf]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1503.04745\"\n     onclick=\"log_download('towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018', 'http://arxiv.org/abs/1503.04745', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>7 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-aninverseackermannianlowerboundonthelocalunconditionalityconstantofthejamesspace-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{towsner15:lower_bound_james_space,\n   author = {{Towsner}, H.},\n    title = &quot;{An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space}&quot;,\njournal={Houston Journal of Mathematics},\n     year = 2018,\n    month = aug,\n    volume = 44,\n    issue = 3,\n    pages = {873--885},\n    urljournal={https://www.math.uh.edu/~hjm/restricted/pdf44(3)/09towsner.pdf},\n    urlarxiv={http://arxiv.org/abs/1503.04745},\n    abstract={The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1204.1884\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018', 'http://arxiv.org/abs/1204.1884', event);\">\n    An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Discrete Analysis</i>, (4).  2018.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1204.1884\"\n     onclick=\"log_download('towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018', 'http://arxiv.org/abs/1204.1884', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal\"\n     onclick=\"log_download('towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018', 'http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>16 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-ananalyticapproachtosparsehypergraphshypergraphremoval-2018')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{towsner12:_analy_approac_spars_hyper,\n  author = \t {Towsner, Henry},\n  title = \t {An Analytic Approach to Sparse Hypergraphs: Hypergraph Removal},\n  year = \t 2018,\n  journal={Discrete Analysis},\n  number={4},\nurlarxiv={http://arxiv.org/abs/1204.1884},\nurljournal={http://discreteanalysisjournal.com/article/3104-an-analytic-approach-to-sparse-hypergraphs-hypergraph-removal},\nabstract={The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.},\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2017\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2017')\"\n      onkeypress=\"toggleGroup('group_2017')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2017</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-moreorlessuniformconvergence-2017\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1709.05687\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-moreorlessuniformconvergence-2017', 'https://arxiv.org/abs/1709.05687', event);\">\n    More or Less Uniform Convergence.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  September 2017.\n  <span class=\"note\">submitted</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1709.05687\"\n     onclick=\"log_download('towsner-moreorlessuniformconvergence-2017', 'https://arxiv.org/abs/1709.05687', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"More or Less Uniform Convergence [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-moreorlessuniformconvergence-2017\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>7 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-moreorlessuniformconvergence-2017')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{1709.05687,\nauthor ={Henry Towsner},\ntitle = {More or Less Uniform Convergence},\nyear={2017},\nmonth= sep,\nnote={submitted},\nurlarxiv={https://arxiv.org/abs/1709.05687},\nabstract={Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals.\n\nWe give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract alpha-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. \n\nThe proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when &quot;proof mining&quot; convergence proofs to obtain quantitative bounds. \n\nWe show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha&lt;omega_1.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals. We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract alpha-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when \"proof mining\" convergence proofs to obtain quantitative bounds. We show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://link.springer.com/article/10.1007/s00153-016-0519-2\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017', 'http://link.springer.com/article/10.1007/s00153-016-0519-2', event);\">\n    Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Igusa, G.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Archive for Mathematical Logic</i>, 56(1).  2017.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://link.springer.com/article/10.1007/s00153-016-0519-2\"\n     onclick=\"log_download('igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017', 'http://link.springer.com/article/10.1007/s00153-016-0519-2', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Computable Ramsey&#x27;s Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1507.03256\"\n     onclick=\"log_download('igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017', 'http://arxiv.org/abs/1507.03256', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Computable Ramsey&#x27;s Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-016-0519-2\"\n     onclick=\"log_download('igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017', 'http://doi.org/10.1007/s00153-016-0519-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('igusa-towsner-computableramseystheoremforpairsneedsinfinitelymanypi02sets-2017')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@ARTICLE{igusa15:computable_ramseys_theorem,\n   author = {{Igusa}, G. and {Towsner}, H.},\n    title = &quot;{Computable Ramsey&#x27;s Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets}&quot;,\nyear=&quot;2017&quot;,\nvolume=&quot;56&quot;,\nnumber=&quot;1&quot;,\n    journal = {Archive for Mathematical Logic},\n    doi=&quot;10.1007/s00153-016-0519-2&quot;,\n    urljournal={http://link.springer.com/article/10.1007/s00153-016-0519-2},\nurlarxiv={http://arxiv.org/abs/1507.03256},\nabstract={In [J], Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite &amp;Pi;&lt;sup&gt;0&lt;/sup&gt;&lt;sup&gt;2&lt;/sub&gt; homogeneous set. The proof uses a countable collection of &amp;Pi;&lt;sup&gt;0&lt;/sup&gt;&lt;sup&gt;2&lt;/sub&gt; sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of &amp;Pi;&lt;sup&gt;0&lt;/sup&gt;&lt;sup&gt;2&lt;/sub&gt; sets. We provide a proof of this latter fact.\n}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In [J], Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite &Pi;<sup>0</sup><sup>2</sub> homogeneous set. The proof uses a countable collection of &Pi;<sup>0</sup><sup>2</sub> sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of &Pi;<sup>0</sup><sup>2</sub> sets. We provide a proof of this latter fact. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-sigmaalgebrasforquasirandomhypergraphs-2017\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-sigmaalgebrasforquasirandomhypergraphs-2017', 'http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract', event);\">\n    Sigma-Algebras for Quasirandom Hypergraphs.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Random Struct. Alg.</i>, 50(1): 114–139.  2017.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract\"\n     onclick=\"log_download('towsner-sigmaalgebrasforquasirandomhypergraphs-2017', 'http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Sigma-Algebras for Quasirandom Hypergraphs [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1312.4882\"\n     onclick=\"log_download('towsner-sigmaalgebrasforquasirandomhypergraphs-2017', 'http://arxiv.org/abs/1312.4882', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Sigma-Algebras for Quasirandom Hypergraphs [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1002/rsa.20641\"\n     onclick=\"log_download('towsner-sigmaalgebrasforquasirandomhypergraphs-2017', 'http://doi.org/10.1002/rsa.20641', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-sigmaalgebrasforquasirandomhypergraphs-2017\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>11 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-sigmaalgebrasforquasirandomhypergraphs-2017')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{towsner13:quasirandom_hypergraphs,\nauthor = {{Towsner}, H.},\ntitle = {Sigma-Algebras for Quasirandom Hypergraphs},\njournal = {Random Struct. Alg.},\nfjournal = {Random Structures \\&amp; Algoritmhs},\nvolume = {50},\nnumber = {1},\npages = {114--139},\nyear = {2017},\nurljournal={http://onlinelibrary.wiley.com/doi/10.1002/rsa.20641/abstract},\ndoi={10.1002/rsa.20641},\nurlarxiv={http://arxiv.org/abs/1312.4882},\nabstract = {We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. \nWe associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, R\\&quot;odl, and Schacht; Conlon, H\\&#x60;an, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.}\n}\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. We associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2016\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2016')\"\n      onkeypress=\"toggleGroup('group_2016')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2016</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"lindell-towsner-weinstein-hanflocalityandinvariantelementarydefinability-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  Hanf Locality and Invariant Elementary Definability.\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Lindell, S.; <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>;  and Weinstein, S.\n</span>\n\n  \n\n</span>\n\n   2016.\n  <span class=\"note\">submitted</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/lindell-towsner-weinstein-hanflocalityandinvariantelementarydefinability-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('lindell-towsner-weinstein-hanflocalityandinvariantelementarydefinability-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{hanf_locality,\nauthor = {Steven Lindell and Henry Towsner and Scott Weinstein},\ntitle = {Hanf Locality and Invariant Elementary Definability},\nyear = {2016},\nnote = {submitted},\nabstract = {We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical rela tions. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call\nsuch expansions presentations of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect\nto elementary, neighborhood-bounded presentations. The second is a non-uniform version of the Fagin-\nStockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, local presentations.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical rela tions. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call such expansions presentations of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect to elementary, neighborhood-bounded presentations. The second is a non-uniform version of the Fagin- Stockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, local presentations.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"crane-towsner-thestructureofcombinatorialmarkovprocesses-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1603.05954\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'crane-towsner-thestructureofcombinatorialmarkovprocesses-2016', 'http://arxiv.org/abs/1603.05954', event);\">\n    The structure of combinatorial Markov processes.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Crane, H.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2016.\n  <span class=\"note\">submitted</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1603.05954\"\n     onclick=\"log_download('crane-towsner-thestructureofcombinatorialmarkovprocesses-2016', 'http://arxiv.org/abs/1603.05954', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"The structure of combinatorial Markov processes [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/crane-towsner-thestructureofcombinatorialmarkovprocesses-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('crane-towsner-thestructureofcombinatorialmarkovprocesses-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@misc{1603.05954,\nAuthor = {Harry Crane and Henry Towsner},\nTitle = {The structure of combinatorial Markov processes},\nYear = {2016},\nEprint = {arXiv:1603.05954},\nnote={submitted},\nurlarxiv={http://arxiv.org/abs/1603.05954},\nabstract={ Every exchangeable Feller process taking values in a suitably\nnice combinatorial state space can be constructed by a system of\niterated random Lipschitz functions. In discrete time, the construction\nproceeds by iterated application of independent, identically distributed\nfunctions, while in continuous time the random functions occur as the\natoms of a time homogeneous Poisson point process. We further show\nthat every exchangeable Feller process projects to a Feller process in an\nappropriate limit space, akin to the projection of partition-valued processes\ninto the ranked-simplex and graph-valued processes into the space\nof graph limits. Together, our main theorems establish common structural\nfeatures shared by all exchangeable combinatorial Feller processes,\nregardless of the dynamics or resident state space, thereby generalizing\nbehaviors previously observed for exchangeable coalescent and fragmentation\nprocesses as well as other combinatorial stochastic processes. If,\nin addition, an exchangeable Feller process evolves on a state space satisfying\nthe n-disjoint amalgamation property for all n &gt;= 1, then its\njump measure can be decomposed explicitly in the sense of Lévy–Itô–\nKhintchine.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the n-disjoint amalgamation property for all n >= 1, then its jump measure can be decomposed explicitly in the sense of Lévy–Itô– Khintchine.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016', 'https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml', event);\">\n    8.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  Volume 3 of Studies in Weak Arithmetic. Towards an Effective Theory of Absolutely Continuous Measures. Patrick Cégielski, A. E.;  and Kossak, R., editor(s). CSLI,  2016.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml\"\n     onclick=\"log_download('towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016', 'https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Towards an Effective Theory of Absolutely Continuous Measures [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1503.04743\"\n     onclick=\"log_download('towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016', 'http://arxiv.org/abs/1503.04743', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Towards an Effective Theory of Absolutely Continuous Measures [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>8 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-towardsaneffectivetheoryofabsolutelycontinuousmeasures-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@inbook{towsner:abs_cont,\n\tAuthor = {Towsner, Henry},\n\tChapter = {8},\n\tEditor = {Patrick C\\&#x27;egielski, Ali Enayat, and Roman Kossak},\n\tPublisher = {CSLI},\n\tSeries = {Studies in Weak Arithmetic},\n\tTitle = {Towards an Effective Theory of Absolutely Continuous Measures},\n\tVolume = {3},\n\tYear = {2016},\nurljournal={https://web.stanford.edu/group/cslipublications/cslipublications/site/9781575867236.shtml},\nurlarxiv={http://arxiv.org/abs/1503.04743},\n    abstract={We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence L^1 functions. A crucial tool is a one-dimensional version of Szemeredi&#x27;s regularity lemma for L^1 functions.\n}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence L^1 functions. A crucial tool is a one-dimensional version of Szemeredi's regularity lemma for L^1 functions. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"flood-towsner-separatingprinciplesbelowwkl0-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1002/malq.201500001\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'flood-towsner-separatingprinciplesbelowwkl0-2016', 'http://dx.doi.org/10.1002/malq.201500001', event);\">\n    Separating Principles Below WKL0.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Flood, S.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Mathematical Logic Quarterly</i>, 62(6): 507–529. December 2016.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1002/malq.201500001\"\n     onclick=\"log_download('flood-towsner-separatingprinciplesbelowwkl0-2016', 'http://dx.doi.org/10.1002/malq.201500001', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Separating Principles Below WKL0 [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1410.4068\"\n     onclick=\"log_download('flood-towsner-separatingprinciplesbelowwkl0-2016', 'http://arxiv.org/abs/1410.4068', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Separating Principles Below WKL0 [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1002/malq.201500001\"\n     onclick=\"log_download('flood-towsner-separatingprinciplesbelowwkl0-2016', 'http://doi.org/10.1002/malq.201500001', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/flood-towsner-separatingprinciplesbelowwkl0-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('flood-towsner-separatingprinciplesbelowwkl0-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{flood14:separating_principles_below_wkl0,\n   author = {{Flood}, S. and {Towsner}, H.},\n    title = &quot;{Separating Principles Below WKL0}&quot;,\njournal={Mathematical Logic Quarterly},\nyear={2016},\nmonth=dec,\nvolume = {62},\nnumber = {6},\nissn = {1521-3870},\nurljournal = {http://dx.doi.org/10.1002/malq.201500001},\ndoi = {10.1002/malq.201500001},\npages = {507--529},\nurlarxiv={http://arxiv.org/abs/1410.4068},\nabstract={In this paper, we study Ramsey-type Konig&#x27;s Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a &quot;one step&quot; construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). \n\nWe illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL from DNR (which has been shown separately by Bienvenu, Patey, and Schafer).\n}\n}\n\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper, we study Ramsey-type Konig's Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a \"one step\" construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). We illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL from DNR (which has been shown separately by Bienvenu, Patey, and Schafer). \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2015\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2015')\"\n      onkeypress=\"toggleGroup('group_2015')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2015</span>\n      <span class=\"bibbase_group_count\">\n        \n        (5)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1007/s00153-015-0449-4\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015', 'http://dx.doi.org/10.1007/s00153-015-0449-4', event);\">\n    Dividing and weak quasi-dimensions in arbitrary theories.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Goldbring, I.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Archive for Mathematical Logic</i>, 54(7-8): 915-920.  2015.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1007/s00153-015-0449-4\"\n     onclick=\"log_download('goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015', 'http://dx.doi.org/10.1007/s00153-015-0449-4', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Dividing and weak quasi-dimensions in arbitrary theories [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-015-0449-4\"\n     onclick=\"log_download('goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015', 'http://doi.org/10.1007/s00153-015-0449-4', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('goldbring-towsner-dividingandweakquasidimensionsinarbitrarytheories-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{goldbring15:dividing,\nyear={2015},\nissn={0933-5846},\njournal={Archive for Mathematical Logic},\nvolume={54},\nnumber={7-8},\ndoi={10.1007/s00153-015-0449-4},\ntitle={Dividing and weak quasi-dimensions in arbitrary theories},\nurl={http://dx.doi.org/10.1007/s00153-015-0449-4},\npublisher={Springer Berlin Heidelberg},\nkeywords={Pseudofinite structure; Quasidimension; Dividing; 03C20; 03C45},\nauthor={Goldbring, Isaac and Towsner, Henry},\npages={915-920},\nlanguage={English},\nabstract={We show that any countable model of a model complete\ntheory has an elementary extension with a “pseudofinite-like” quasidimension\nthat detects dividing.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasidimension that detects dividing.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-aworkedexampleofthefunctionalinterpretation-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1503.05572\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-aworkedexampleofthefunctionalinterpretation-2015', 'http://arxiv.org/abs/1503.05572', event);\">\n    A Worked Example of the Functional Interpretation.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  March 2015.\n  <span class=\"note\">submitted</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1503.05572\"\n     onclick=\"log_download('towsner-aworkedexampleofthefunctionalinterpretation-2015', 'http://arxiv.org/abs/1503.05572', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Worked Example of the Functional Interpretation [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-aworkedexampleofthefunctionalinterpretation-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>10 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-aworkedexampleofthefunctionalinterpretation-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@unpublished{towsner15:worked_example,\n   author = {{Towsner}, H.},\n    title = &quot;{A Worked Example of the Functional Interpretation}&quot;,\n    note={submitted},\n     year = 2015,\n    month = mar,\n   urlarxiv = {http://arxiv.org/abs/1503.05572},\n  abstract = {The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-limitsofsequencesofmarkovchains-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://ejp.ejpecp.org/article/view/4188\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-limitsofsequencesofmarkovchains-2015', 'http://ejp.ejpecp.org/article/view/4188', event);\">\n    Limits of sequences of Markov chains.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Electron. J. Probab.</i>, 20: no. 77, 1-23.  2015.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://ejp.ejpecp.org/article/view/4188\"\n     onclick=\"log_download('towsner-limitsofsequencesofmarkovchains-2015', 'http://ejp.ejpecp.org/article/view/4188', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Limits of sequences of Markov chains [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1404.3815\"\n     onclick=\"log_download('towsner-limitsofsequencesofmarkovchains-2015', 'http://arxiv.org/abs/1404.3815', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Limits of sequences of Markov chains [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1214/EJP.v20-4188\"\n     onclick=\"log_download('towsner-limitsofsequencesofmarkovchains-2015', 'http://doi.org/10.1214/EJP.v20-4188', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-limitsofsequencesofmarkovchains-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>8 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-limitsofsequencesofmarkovchains-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{towsner14:sequence_markov_chains,\n\tauthor = {Henry Towsner},\n\ttitle = {Limits of sequences of Markov chains},\n\tjournal = {Electron. J. Probab.},\n\tfjournal = {Electronic Journal of Probability},\n\tvolume = {20},\n\tyear = {2015},\n\tkeywords = {Markov chain, graph limit, ultraproduct},\n\tabstract = {We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space.  The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called &quot;random-free&quot; or &quot;product measurable&quot;) is simpler than the general case.},\n\tpages = {no. 77, 1-23},\n\tissn = {1083-6489},\n\tdoi = {10.1214/EJP.v20-4188},    \n        urljournal = {http://ejp.ejpecp.org/article/view/4188},\nurlarxiv={http://arxiv.org/abs/1404.3815},\n}\n\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called \"random-free\" or \"product measurable\") is simpler than the general case.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015', 'http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf', event);\">\n    Infinitary methods in finite model theory.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Lindell, S.; <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>;  and Weinstein, S.\n</span>\n\n  \n\n</span>\n\n  In Hirvonen, A.; Kontinen, J.; Kossak, R.;  and Villaveces, A., editor(s), <i>Logic Without Borders</i>. De Gruyter,  2015.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf\"\n     onclick=\"log_download('lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015', 'http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Infinitary methods in finite model theory [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('lindell-towsner-weinstein-infinitarymethodsinfinitemodeltheory-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@InCollection{lindell13:_infin,\n  author = \t {Steven Lindell and Henry Towsner and Scott Weinstein},\n  title = \t {Infinitary methods in finite model theory},\n  booktitle = {Logic Without Borders},\n  publisher = {de Gruyter},\n  year = \t 2015,\n  editor = \t {Asa Hirvonen and Juha Kontinen and Roman Kossak and Andr\\&#x27;es Villaveces},\n  urlpdf = {http://ww3.haverford.edu/cmsc/slindell/Infinitary%20methods.pdf},\nabstract={\nThis paper explores applications of the infinitary techniques of classical model theory in\nthe context of finite model theory, and thus elaborates on themes explored by V&amp;auml;&amp;auml;n&amp;auml;nen\n(2003). We are grateful for the opportunity to celebrate Jouko&#x27;s inclusive vision of logic\nin this way.\n}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper explores applications of the infinitary techniques of classical model theory in the context of finite model theory, and thus elaborates on themes explored by V&auml;&auml;n&auml;nen (2003). We are grateful for the opportunity to celebrate Jouko's inclusive vision of logic in this way. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-onmaximumconservativeextensions-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1302.1488\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-onmaximumconservativeextensions-2015', 'http://arxiv.org/abs/1302.1488', event);\">\n    On Maximum Conservative Extensions.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Computability</i>, 4: 57–68.  2015.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1302.1488\"\n     onclick=\"log_download('towsner-onmaximumconservativeextensions-2015', 'http://arxiv.org/abs/1302.1488', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On Maximum Conservative Extensions [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-onmaximumconservativeextensions-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>12 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-onmaximumconservativeextensions-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Article{towsner13:maximal_conservatve,\n  author = \t {Towsner, H.},\n  title = \t {On Maximum Conservative Extensions},\n  journal = \t {Computability},\n  year = \t 2015,\n  volume = \t 4,\n  pages = \t {57--68},\nurlarxiv={http://arxiv.org/abs/1302.1488},\nabstract={We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma-n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi-2 complete.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma-n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi-2 complete.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2014\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2014')\"\n      onkeypress=\"toggleGroup('group_2014')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2014</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"franklin-towsner-randomnessandnonergodicsystems-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1206.2682\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'franklin-towsner-randomnessandnonergodicsystems-2014', 'http://arxiv.org/abs/1206.2682', event);\">\n    Randomness and non-ergodic systems.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Franklin, J. N. Y.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Mosc. Math. J.</i>, 14(4): 711–744, 827.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1206.2682\"\n     onclick=\"log_download('franklin-towsner-randomnessandnonergodicsystems-2014', 'http://arxiv.org/abs/1206.2682', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Randomness and non-ergodic systems [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html\"\n     onclick=\"log_download('franklin-towsner-randomnessandnonergodicsystems-2014', 'http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Randomness and non-ergodic systems [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/franklin-towsner-randomnessandnonergodicsystems-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('franklin-towsner-randomnessandnonergodicsystems-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {franklin12:_random_non_system,\n    AUTHOR = {Franklin, Johanna N. Y. and Towsner, Henry},\n     TITLE = {Randomness and non-ergodic systems},\n   JOURNAL = {Mosc. Math. J.},\n  FJOURNAL = {Moscow Mathematical Journal},\n    VOLUME = {14},\n      YEAR = {2014},\n    NUMBER = {4},\n     PAGES = {711--744, 827},\n      ISSN = {1609-3321},\n   MRCLASS = {37A25 (03D32)},\n  MRNUMBER = {3292047},\nurlarxiv={http://arxiv.org/abs/1206.2682},\nurljournal={http://www.mathjournals.org/mmj/2014-014-004/2014-014-004-004.html},\nabstract={We characterize the points that satisfy Birkhoff&#x27;s ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V&#x27;yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.},\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-ultrafiltersinreversemathematics-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/1109.3902\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-ultrafiltersinreversemathematics-2014', 'http://arxiv.org/abs/1109.3902', event);\">\n    Ultrafilters in reverse mathematics.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>J. Math. Logic</i>, 14(01): 1450001.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/1109.3902\"\n     onclick=\"log_download('towsner-ultrafiltersinreversemathematics-2014', 'http://arxiv.org/abs/1109.3902', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Ultrafilters in reverse mathematics [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019\"\n     onclick=\"log_download('towsner-ultrafiltersinreversemathematics-2014', 'http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Ultrafilters in reverse mathematics [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1142/S0219061314500019\"\n     onclick=\"log_download('towsner-ultrafiltersinreversemathematics-2014', 'http://doi.org/10.1142/S0219061314500019', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-ultrafiltersinreversemathematics-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>155 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-ultrafiltersinreversemathematics-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{doi:10.1142/S0219061314500019,\nauthor = {Towsner, Henry},\ntitle = {Ultrafilters in reverse mathematics},\njournal={J. Math. Logic},\nfjournal={Journal of Mathematical Logic},\nvolume = {14},\nnumber = {01},\npages = {1450001},\nyear = {2014},\ndoi = {10.1142/S0219061314500019},\nurlarxiv={http://arxiv.org/abs/1109.3902},\nabstract={We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA&lt;sub&gt;0&lt;/sub&gt;, ATR&lt;sub&gt;0&lt;/sub&gt;, and &amp;Pi;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;1&lt;/sub&gt;-Comprehension.},\nurljournal = {http://www.worldscientific.com/doi/pdf/10.1142/S0219061314500019}\n}\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA<sub>0</sub>, ATR<sub>0</sub>, and &Pi;<sup>1</sup><sub>1</sub>-Comprehension.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"goldbring-towsner-anapproximatelogicformeasures-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1007/s11856-013-0054-3\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'goldbring-towsner-anapproximatelogicformeasures-2014', 'http://dx.doi.org/10.1007/s11856-013-0054-3', event);\">\n    An approximate logic for measures.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Goldbring, I.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Israel J. Math.</i>, 199(2): 867-913.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1007/s11856-013-0054-3\"\n     onclick=\"log_download('goldbring-towsner-anapproximatelogicformeasures-2014', 'http://dx.doi.org/10.1007/s11856-013-0054-3', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An approximate logic for measures [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1106.2854\"\n     onclick=\"log_download('goldbring-towsner-anapproximatelogicformeasures-2014', 'http://arxiv.org/abs/1106.2854', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"An approximate logic for measures [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s11856-013-0054-3\"\n     onclick=\"log_download('goldbring-towsner-anapproximatelogicformeasures-2014', 'http://doi.org/10.1007/s11856-013-0054-3', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/goldbring-towsner-anapproximatelogicformeasures-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>69 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('goldbring-towsner-anapproximatelogicformeasures-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{goldbring:_approx_logic_measur,\nyear={2014},\nissn={0021-2172},\njournal={Israel J. Math.},\nfjournal={Israel Journal of Mathematics},\nvolume={199},\nnumber={2},\ndoi={10.1007/s11856-013-0054-3},\ntitle={An approximate logic for measures},\nurljournal={http://dx.doi.org/10.1007/s11856-013-0054-3},\npublisher={The Hebrew University Magnes Press},\nauthor={Goldbring, Isaac and Towsner, Henry},\npages={867-913},\nlanguage={English},\nurlarxiv={http://arxiv.org/abs/1106.2854},\nabstract={We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer&amp;eacute;di Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.},\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer&eacute;di Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2013\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2013')\"\n      onkeypress=\"toggleGroup('group_2013')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2013</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013', 'http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074', event);\">\n    Separating principles below Ramsey's Theorem for Pairs.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Lerman, M.; Solomon, R.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>J. Math. Logic</i>, 13(02): 1350007.  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074\"\n     onclick=\"log_download('lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013', 'http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Separating principles below Ramsey&#x27;s Theorem for Pairs [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1302.0828\"\n     onclick=\"log_download('lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013', 'http://arxiv.org/abs/1302.0828', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Separating principles below Ramsey&#x27;s Theorem for Pairs [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1142/S0219061313500074\"\n     onclick=\"log_download('lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013', 'http://doi.org/10.1142/S0219061313500074', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>35 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('lerman-solomon-towsner-separatingprinciplesbelowramseystheoremforpairs-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{doi:10.1142/S0219061313500074,\nauthor = {{Lerman}, M. and {Solomon}, R. and {Towsner}, H.},\ntitle = {Separating principles below Ramsey&#x27;s Theorem for Pairs},\njournal={J. Math. Logic},\nfjournal={Journal of Mathematical Logic},\nvolume = {13},\nnumber = {02},\npages = {1350007},\nyear = {2013},\ndoi = {10.1142/S0219061313500074},\nURLjournal = {http://www.worldscientific.com/doi/abs/10.1142/S0219061313500074},\nurlarxiv={http://arxiv.org/abs/1302.0828},\nabstract={In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey&#x27;s Theorem for Pairs. These principles tend to fall outside of the &quot;big five&quot; systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.},\n}\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the \"big five\" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-partialimpredicativityinreversemathematics-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://projecteuclid.org/euclid.jsl/1368627060\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-partialimpredicativityinreversemathematics-2013', 'http://projecteuclid.org/euclid.jsl/1368627060', event);\">\n    Partial Impredicativity in Reverse Mathematics.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>J. Symbolic Logic</i>, 78(2): 459–488.  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://projecteuclid.org/euclid.jsl/1368627060\"\n     onclick=\"log_download('towsner-partialimpredicativityinreversemathematics-2013', 'http://projecteuclid.org/euclid.jsl/1368627060', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Partial Impredicativity in Reverse Mathematics [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1106.6063\"\n     onclick=\"log_download('towsner-partialimpredicativityinreversemathematics-2013', 'http://arxiv.org/abs/1106.6063', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Partial Impredicativity in Reverse Mathematics [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-partialimpredicativityinreversemathematics-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>33 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-partialimpredicativityinreversemathematics-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{towsner11:_approx_impred_rever_mathem,\n  author = \t {Towsner, Henry},\n  title = \t {Partial Impredicativity in Reverse Mathematics},\njournal={J. Symbolic Logic},\nfjournal={Journal of Symbolic Logic},\nvolume={78},\nnumber={2},\npages={459--488},\nurljournal={http://projecteuclid.org/euclid.jsl/1368627060},\n  year = \t 2013,\nurlarxiv={http://arxiv.org/abs/1106.6063},\nabstract={In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a &amp;Pi;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;2&lt;/sub&gt; sentence from the theory &amp;Pi;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;1&lt;/sub&gt;-Comprehension. Using methods based on the functional interpretation, we introduce a family of weakenings of &amp;Pi;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;1&lt;/sub&gt;-Comprehension and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger&#x27;s Theorem for countable graphs.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a &Pi;<sup>1</sup><sub>2</sub> sentence from the theory &Pi;<sup>1</sup><sub>1</sub>-Comprehension. Using methods based on the functional interpretation, we introduce a family of weakenings of &Pi;<sup>1</sup><sub>1</sub>-Comprehension and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2012\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2012')\"\n      onkeypress=\"toggleGroup('group_2012')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2012</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1007/s11856-011-0195-1\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012', 'http://dx.doi.org/10.1007/s11856-011-0195-1', event);\">\n    Transfinite approximation of Hindman's theorem.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Beiglbock, M.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Israel J. Math.</i>, 191: 41–59.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1007/s11856-011-0195-1\"\n     onclick=\"log_download('beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012', 'http://dx.doi.org/10.1007/s11856-011-0195-1', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Transfinite approximation of Hindman&#x27;s theorem [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1001.1175\"\n     onclick=\"log_download('beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012', 'http://arxiv.org/abs/1001.1175', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Transfinite approximation of Hindman&#x27;s theorem [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s11856-011-0195-1\"\n     onclick=\"log_download('beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012', 'http://doi.org/10.1007/s11856-011-0195-1', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>45 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('beiglbock-towsner-transfiniteapproximationofhindmanstheorem-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2970862,\n    AUTHOR = {Beiglbock, Mathias and Towsner, Henry},\n     TITLE = {Transfinite approximation of {H}indman&#x27;s theorem},\n   JOURNAL = {Israel J. Math.},\n  FJOURNAL = {Israel Journal of Mathematics},\n    VOLUME = {191},\n      YEAR = {2012},\n     PAGES = {41--59},\n      ISSN = {0021-2172},\n     CODEN = {ISJMAP},\n   MRCLASS = {Preliminary Data},\n  MRNUMBER = {2970862},\n       DOI = {10.1007/s11856-011-0195-1},\n       URLJOURNAL = {http://dx.doi.org/10.1007/s11856-011-0195-1},\nurlarxiv={http://arxiv.org/abs/1001.1175},\nabstract={Hindman&#x27;s Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman&#x27;s Theorem to a &quot;transfinite&quot; version for each countable ordinal, and show that Hindman&#x27;s Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman&#x27;s Theorem by directly proving these transfinite approximations.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a \"transfinite\" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://projecteuclid.org/euclid.ndjfl/1336586237\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012', 'http://projecteuclid.org/euclid.ndjfl/1336586237', event);\">\n    A Simple Proof and Some Difficult Examples for Hindman's Theorem.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Notre Dame J. Formal Logic</i>, 53(1): 53-65.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://projecteuclid.org/euclid.ndjfl/1336586237\"\n     onclick=\"log_download('towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012', 'http://projecteuclid.org/euclid.ndjfl/1336586237', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Simple Proof and Some Difficult Examples for Hindman&#x27;s Theorem [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0906.3885\"\n     onclick=\"log_download('towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012', 'http://arxiv.org/abs/0906.3885', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Simple Proof and Some Difficult Examples for Hindman&#x27;s Theorem [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>47 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-asimpleproofandsomedifficultexamplesforhindmanstheorem-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Article{towsner12:_simpl_proof_some_diffic_examp_hindm_theor,\n  author = \t {Towsner, Henry},\n  title = \t {A Simple Proof and Some Difficult Examples for {H}indman&#x27;s Theorem},\njournal = {Notre Dame J. Formal Logic},\n  fjournal = \t {Notre Dame Journal of Formal Logic},\n  year = \t 2012,\n  volume = \t 53,\n  number = \t 1,\n  pages = \t {53-65},\nurljournal={http://projecteuclid.org/euclid.ndjfl/1336586237},\nurlarxiv={http://arxiv.org/abs/0906.3885},\nabstract={We give a short, explicit proof of Hindman&#x27;s Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman&#x27;s Theorem.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://logicandanalysis.org/index.php/jla/article/view/139/51\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012', 'http://logicandanalysis.org/index.php/jla/article/view/139/51', event);\">\n    A Correspondence Principle for the Gowers Uniformity Norms.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Journal of Logic and Analysis</i>, 4. February 2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://logicandanalysis.org/index.php/jla/article/view/139/51\"\n     onclick=\"log_download('towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012', 'http://logicandanalysis.org/index.php/jla/article/view/139/51', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Correspondence Principle for the Gowers Uniformity Norms [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0905.0493\"\n     onclick=\"log_download('towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012', 'http://arxiv.org/abs/0905.0493', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A Correspondence Principle for the Gowers Uniformity Norms [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>30 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-acorrespondenceprincipleforthegowersuniformitynorms-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Article{towsner12:_corres_princ_gower_unifor_norms,\n  author = \t {Towsner, Henry},\n  title = \t {A Correspondence Principle for the {G}owers Uniformity Norms},\n  journal = \t {Journal of Logic and Analysis},\n  year = \t 2012,\n  volume = \t 4,\n  month = \t {February},\nurljournal={http://logicandanalysis.org/index.php/jla/article/view/139/51},\nurlarxiv={http://arxiv.org/abs/0905.0493},\nabstract={We give a variation of the Furstenberg Correspondence which preserves the Gowers uniformity norms.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We give a variation of the Furstenberg Correspondence which preserves the Gowers uniformity norms.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2011\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2011')\"\n      onkeypress=\"toggleGroup('group_2011')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2011</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-acombinatorialproofofthedensehindmanstheorem-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1016/j.disc.2011.03.006\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-acombinatorialproofofthedensehindmanstheorem-2011', 'http://dx.doi.org/10.1016/j.disc.2011.03.006', event);\">\n    A combinatorial proof of the dense Hindman's theorem.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Discrete Math.</i>, 311(14): 1380–1384.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1016/j.disc.2011.03.006\"\n     onclick=\"log_download('towsner-acombinatorialproofofthedensehindmanstheorem-2011', 'http://dx.doi.org/10.1016/j.disc.2011.03.006', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A combinatorial proof of the dense Hindman&#x27;s theorem [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/1002.0347\"\n     onclick=\"log_download('towsner-acombinatorialproofofthedensehindmanstheorem-2011', 'http://arxiv.org/abs/1002.0347', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A combinatorial proof of the dense Hindman&#x27;s theorem [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.disc.2011.03.006\"\n     onclick=\"log_download('towsner-acombinatorialproofofthedensehindmanstheorem-2011', 'http://doi.org/10.1016/j.disc.2011.03.006', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-acombinatorialproofofthedensehindmanstheorem-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>44 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-acombinatorialproofofthedensehindmanstheorem-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2795548,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {A combinatorial proof of the dense {H}indman&#x27;s theorem},\n   JOURNAL = {Discrete Math.},\n  FJOURNAL = {Discrete Mathematics},\n    VOLUME = {311},\n      YEAR = {2011},\n    NUMBER = {14},\n     PAGES = {1380--1384},\n      ISSN = {0012-365X},\n     CODEN = {DSMHA4},\n   MRCLASS = {05D10 (03C20)},\n  MRNUMBER = {2795548 (2012h:05349)},\nMRREVIEWER = {Damir D. Dzhafarov},\n       DOI = {10.1016/j.disc.2011.03.006},\n       URLJOURNAL= {http://dx.doi.org/10.1016/j.disc.2011.03.006},\nurlarxiv={http://arxiv.org/abs/1002.0347},\nabstract={The Dense Hindman&#x27;s Theorem states that, in any finite coloring of the integers, one may find a single color and a &quot;dense&quot; set B&lt;sub&gt;1&lt;/sub&gt;, for each b&lt;sub&gt;1&lt;/sub&gt;&amp;isin; B&lt;sub&gt;1&lt;/sub&gt; a &quot;dense&quot; set B&lt;sub&gt;2&lt;/sub&gt;(b&lt;sub&gt;1&lt;/sub&gt;) (depending on b&lt;sub&gt;1&lt;/sub&gt;), for each b&lt;sub&gt;2&lt;/sub&gt;&amp;isin; B&lt;sub&gt;2&lt;/sub&gt;(b&lt;sub&gt;1&lt;/sub&gt;) a &quot;dense&quot; set B&lt;sub&gt;3&lt;/sub&gt;(b&lt;sub&gt;1&lt;/sub&gt;, b&lt;sub&gt;2&lt;/sub&gt;) (depending on b&lt;sub&gt;1&lt;/sub&gt;, b&lt;sub&gt;2&lt;/sub&gt;), and so on, such that for any such sequence of b&lt;sub&gt;i&lt;/sub&gt;, all finite sums belong to the chosen color. (Here density is often taken to be &quot;piecewise syndetic&quot;, but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.}\n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a \"dense\" set B<sub>1</sub>, for each b<sub>1</sub>&isin; B<sub>1</sub> a \"dense\" set B<sub>2</sub>(b<sub>1</sub>) (depending on b<sub>1</sub>), for each b<sub>2</sub>&isin; B<sub>2</sub>(b<sub>1</sub>) a \"dense\" set B<sub>3</sub>(b<sub>1</sub>, b<sub>2</sub>) (depending on b<sub>1</sub>, b<sub>2</sub>), and so on, such that for any such sequence of b<sub>i</sub>, all finite sums belong to the chosen color. (Here density is often taken to be \"piecewise syndetic\", but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.2178/jsl/1294171005\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011', 'http://dx.doi.org/10.2178/jsl/1294171005', event);\">\n    Hindman's theorem: an ultrafilter argument in second order arithmetic.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>J. Symbolic Logic</i>, 76(1): 353–360.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.2178/jsl/1294171005\"\n     onclick=\"log_download('towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011', 'http://dx.doi.org/10.2178/jsl/1294171005', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Hindman&#x27;s theorem: an ultrafilter argument in second order arithmetic [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0906.3882\"\n     onclick=\"log_download('towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011', 'http://arxiv.org/abs/0906.3882', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Hindman&#x27;s theorem: an ultrafilter argument in second order arithmetic [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.2178/jsl/1294171005\"\n     onclick=\"log_download('towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011', 'http://doi.org/10.2178/jsl/1294171005', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>56 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-hindmanstheoremanultrafilterargumentinsecondorderarithmetic-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2791353,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {Hindman&#x27;s theorem: an ultrafilter argument in second order\n              arithmetic},\n   JOURNAL = {J. Symbolic Logic},\n  FJOURNAL = {Journal of Symbolic Logic},\n    VOLUME = {76},\n      YEAR = {2011},\n    NUMBER = {1},\n     PAGES = {353--360},\n      ISSN = {0022-4812},\n     CODEN = {JSYLA6},\n   MRCLASS = {03F35 (54D80)},\n  MRNUMBER = {2791353 (2012b:03165)},\nMRREVIEWER = {M. Yasuhara},\n       DOI = {10.2178/jsl/1294171005},\n       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1294171005},\nurlarxiv={http://arxiv.org/abs/0906.3882},\nabstract={Hindman&#x27;s Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.},\n}\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2010\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2010')\"\n      onkeypress=\"toggleGroup('group_2010')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2010</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"avigad-towsner-metastabilityinthefurstenbergzimmertower-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.4064/fm210-3-2\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'avigad-towsner-metastabilityinthefurstenbergzimmertower-2010', 'http://dx.doi.org/10.4064/fm210-3-2', event);\">\n    Metastability in the Furstenberg-Zimmer tower.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Avigad, J.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Fund. Math.</i>, 210(3): 243–268.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.4064/fm210-3-2\"\n     onclick=\"log_download('avigad-towsner-metastabilityinthefurstenbergzimmertower-2010', 'http://dx.doi.org/10.4064/fm210-3-2', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Metastability in the Furstenberg-Zimmer tower [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0902.0356\"\n     onclick=\"log_download('avigad-towsner-metastabilityinthefurstenbergzimmertower-2010', 'http://arxiv.org/abs/0902.0356', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Metastability in the Furstenberg-Zimmer tower [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.4064/fm210-3-2\"\n     onclick=\"log_download('avigad-towsner-metastabilityinthefurstenbergzimmertower-2010', 'http://doi.org/10.4064/fm210-3-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/avigad-towsner-metastabilityinthefurstenbergzimmertower-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>23 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('avigad-towsner-metastabilityinthefurstenbergzimmertower-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2733051,\n    AUTHOR = {Avigad, Jeremy and Towsner, Henry},\n     TITLE = {Metastability in the {F}urstenberg-{Z}immer tower},\n   JOURNAL = {Fund. Math.},\n  FJOURNAL = {Fundamenta Mathematicae},\n    VOLUME = {210},\n      YEAR = {2010},\n    NUMBER = {3},\n     PAGES = {243--268},\n      ISSN = {0016-2736},\n   MRCLASS = {37A05 (03E15 03F03 28D05 37A25)},\n  MRNUMBER = {2733051 (2012b:37003)},\nMRREVIEWER = {Michael H. Schraudner},\n       DOI = {10.4064/fm210-3-2},\n       URLJOURNAL= {http://dx.doi.org/10.4064/fm210-3-2},\nurlarxiv={http://arxiv.org/abs/0902.0356},\nabstract={According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer&amp;eacute;di&#x27;s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the &amp;\\omega;&lt;sup&gt;&amp;\\omega;&lt;sup&gt;&amp;\\omega;&lt;/sup&gt;&lt;/sup&gt;th level.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer&eacute;di's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the &ω;<sup>&ω;<sup>&ω;</sup></sup>th level.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-priorityargumentsandepsilonsubstitution-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/0812.3434\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-priorityargumentsandepsilonsubstitution-2010', 'http://arxiv.org/abs/0812.3434', event);\">\n    Priority arguments and epsilon substitution.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  In <i>Proofs, categories and computations</i>, volume 13, of Tributes, pages 251–266. Coll. Publ., London,  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/0812.3434\"\n     onclick=\"log_download('towsner-priorityargumentsandepsilonsubstitution-2010', 'http://arxiv.org/abs/0812.3434', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Priority arguments and epsilon substitution [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-priorityargumentsandepsilonsubstitution-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>24 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-priorityargumentsandepsilonsubstitution-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@incollection {MR2856914,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {Priority arguments and epsilon substitution},\n BOOKTITLE = {Proofs, categories and computations},\n    SERIES = {Tributes},\n    VOLUME = {13},\n     PAGES = {251--266},\n PUBLISHER = {Coll. Publ., London},\n      YEAR = {2010},\n   MRCLASS = {03F30 (03D25 03D30 03F03)},\n  MRNUMBER = {2856914 (2012i:03165)},\nurlarxiv={http://arxiv.org/abs/0812.3434},\nabstract={Kreisel has observed that the termination proof for Hilbert&#x27;s epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"avigad-gerhardy-towsner-localstabilityofergodicaverages-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1090/S0002-9947-09-04814-4\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'avigad-gerhardy-towsner-localstabilityofergodicaverages-2010', 'http://dx.doi.org/10.1090/S0002-9947-09-04814-4', event);\">\n    Local stability of ergodic averages.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Avigad, J.; Gerhardy, P.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Trans. Amer. Math. Soc.</i>, 362(1): 261–288.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1090/S0002-9947-09-04814-4\"\n     onclick=\"log_download('avigad-gerhardy-towsner-localstabilityofergodicaverages-2010', 'http://dx.doi.org/10.1090/S0002-9947-09-04814-4', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Local stability of ergodic averages [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0706.1512\"\n     onclick=\"log_download('avigad-gerhardy-towsner-localstabilityofergodicaverages-2010', 'http://arxiv.org/abs/0706.1512', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Local stability of ergodic averages [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1090/S0002-9947-09-04814-4\"\n     onclick=\"log_download('avigad-gerhardy-towsner-localstabilityofergodicaverages-2010', 'http://doi.org/10.1090/S0002-9947-09-04814-4', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/avigad-gerhardy-towsner-localstabilityofergodicaverages-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>23 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('avigad-gerhardy-towsner-localstabilityofergodicaverages-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2550151,\n    AUTHOR = {Avigad, Jeremy and Gerhardy, Philipp and Towsner, Henry},\n     TITLE = {Local stability of ergodic averages},\n   JOURNAL = {Trans. Amer. Math. Soc.},\n  FJOURNAL = {Transactions of the American Mathematical Society},\n    VOLUME = {362},\n      YEAR = {2010},\n    NUMBER = {1},\n     PAGES = {261--288},\n      ISSN = {0002-9947},\n     CODEN = {TAMTAM},\n   MRCLASS = {03F60 (28D05)},\n  MRNUMBER = {2550151 (2011e:03082)},\nMRREVIEWER = {Josef Berger},\n       DOI = {10.1090/S0002-9947-09-04814-4},\n       URLJOURNAL= {http://dx.doi.org/10.1090/S0002-9947-09-04814-4},\nurlarxiv={http://arxiv.org/abs/0706.1512},\nabstract={The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A&lt;sub&gt;m&lt;/sub&gt; f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are &quot;locally stable&quot; on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of &quot;proof mining.&quot;},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A<sub>m</sub> f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are \"locally stable\" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of \"proof mining.\"\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2009\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2009')\"\n      onkeypress=\"toggleGroup('group_2009')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2009</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://arxiv.org/abs/0909.5668\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009', 'http://arxiv.org/abs/0909.5668', event);\">\n    Metastability in the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemerédi's Theorem.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n   2009.\n  <span class=\"note\">draft</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://arxiv.org/abs/0909.5668\"\n     onclick=\"log_download('towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009', 'http://arxiv.org/abs/0909.5668', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Metastability in the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemerédi&#x27;s Theorem [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-metastabilityinthefurstenbergzimmertoweriipolynomialandmultidimensionalszemerdistheorem-2009')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Unpublished{towsner09:_metas_furst_zimmer_tower_ii,\n  author = \t {Towsner, Henry},\n  title = \t {Metastability in the {F}urstenberg-{Z}immer Tower II: Polynomial and Multidimensional {S}zemer\\&#x27;edi&#x27;s Theorem},\n  note = \t {draft},\n  year = \t 2009,\nurlarxiv={http://arxiv.org/abs/0909.5668},\nabstract={The Furstenberg-Zimmer structure theorem for &amp;integers;&lt;sub&gt;d,&lt;/sub&gt; actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemer\\&#x27;edi&#x27;s theorem, and Bergelson and Liebman further generalized to a polynomial Szemer\\&#x27;edi&#x27;s theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, &amp;omega;&lt;sup&gt;&amp;omega;&lt;sup&gt;&amp;omega;&lt;sup&gt;&amp;omega;&lt;/sup&gt;&lt;/sup&gt;&lt;/sup&gt;).\n}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Furstenberg-Zimmer structure theorem for &integers;<sub>d,</sub> actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, &omega;<sup>&omega;<sup>&omega;<sup>&omega;</sup></sup></sup>). \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"avigad-towsner-functionalinterpretationandinductivedefinitions-2009\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.2178/jsl/1254748682\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'avigad-towsner-functionalinterpretationandinductivedefinitions-2009', 'http://dx.doi.org/10.2178/jsl/1254748682', event);\">\n    Functional interpretation and inductive definitions.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  Avigad, J.;  and <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>J. Symbolic Logic</i>, 74(4): 1100–1120.  2009.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.2178/jsl/1254748682\"\n     onclick=\"log_download('avigad-towsner-functionalinterpretationandinductivedefinitions-2009', 'http://dx.doi.org/10.2178/jsl/1254748682', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Functional interpretation and inductive definitions [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0802.1938\"\n     onclick=\"log_download('avigad-towsner-functionalinterpretationandinductivedefinitions-2009', 'http://arxiv.org/abs/0802.1938', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Functional interpretation and inductive definitions [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.2178/jsl/1254748682\"\n     onclick=\"log_download('avigad-towsner-functionalinterpretationandinductivedefinitions-2009', 'http://doi.org/10.2178/jsl/1254748682', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/avigad-towsner-functionalinterpretationandinductivedefinitions-2009\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>35 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('avigad-towsner-functionalinterpretationandinductivedefinitions-2009')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2583811,\n    AUTHOR = {Avigad, Jeremy and Towsner, Henry},\n     TITLE = {Functional interpretation and inductive definitions},\n   JOURNAL = {J. Symbolic Logic},\n  FJOURNAL = {Journal of Symbolic Logic},\n    VOLUME = {74},\n      YEAR = {2009},\n    NUMBER = {4},\n     PAGES = {1100--1120},\n      ISSN = {0022-4812},\n     CODEN = {JSYLA6},\n   MRCLASS = {03F10 (03F15 03F35)},\n  MRNUMBER = {2583811 (2011c:03137)},\nMRREVIEWER = {H. Schwichtenberg},\n       DOI = {10.2178/jsl/1254748682},\n       URLJOURNAL= {http://dx.doi.org/10.2178/jsl/1254748682},\nurlarxiv={http://arxiv.org/abs/0802.1938},\nabstract={Extending G&amp;ouml;odel&#x27;s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Extending G&ouml;odel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-convergenceofdiagonalergodicaverages-2009\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://dx.doi.org/10.1017/S0143385708000722\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-convergenceofdiagonalergodicaverages-2009', 'http://dx.doi.org/10.1017/S0143385708000722', event);\">\n    Convergence of diagonal ergodic averages.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Ergodic Theory Dynam. Systems</i>, 29(4): 1309–1326.  2009.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1017/S0143385708000722\"\n     onclick=\"log_download('towsner-convergenceofdiagonalergodicaverages-2009', 'http://dx.doi.org/10.1017/S0143385708000722', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Convergence of diagonal ergodic averages [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://arxiv.org/abs/0711.1180\"\n     onclick=\"log_download('towsner-convergenceofdiagonalergodicaverages-2009', 'http://arxiv.org/abs/0711.1180', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Convergence of diagonal ergodic averages [link]\"\n       title=\"Arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1017/S0143385708000722\"\n     onclick=\"log_download('towsner-convergenceofdiagonalergodicaverages-2009', 'http://doi.org/10.1017/S0143385708000722', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-convergenceofdiagonalergodicaverages-2009\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>21 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-convergenceofdiagonalergodicaverages-2009')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2529651,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {Convergence of diagonal ergodic averages},\n   JOURNAL = {Ergodic Theory Dynam. Systems},\n  FJOURNAL = {Ergodic Theory and Dynamical Systems},\n    VOLUME = {29},\n      YEAR = {2009},\n    NUMBER = {4},\n     PAGES = {1309--1326},\n      ISSN = {0143-3857},\n   MRCLASS = {37A30 (28D15)},\n  MRNUMBER = {2529651 (2010f:37005)},\nMRREVIEWER = {Bryna Kra},\n       DOI = {10.1017/S0143385708000722},\n       URLJOURNAL= {http://dx.doi.org/10.1017/S0143385708000722},\nurlarxiv={http://arxiv.org/abs/0711.1180},\nabstract={Tao has recently proved that if T&lt;sub&gt;1&lt;/sub&gt;,&amp;hellip;,T&lt;sub&gt;l&lt;/sub&gt; are commuting, invertible, measure-preserving transformations on a dynamical system then for any L&lt;sub&gt;&amp;infin;&lt;/sub&gt; functions f&lt;sub&gt;1&lt;/sub&gt;,&amp;hellip;f&lt;sub&gt;l&lt;/sub&gt;, the average (1/N)&amp;Sigma;&lt;sub&gt;n=0&lt;/sub&gt;&lt;sup&gt;N-1&lt;/sup&gt;&amp;Pi;&lt;sub&gt;i&amp;leq;l&lt;/sub&gt;f&lt;sub&gt;i&lt;/sub&gt; o T&lt;sup&gt;in&lt;/sup&gt; converges in the L&lt;sub&gt;2&lt;/sub&gt; norm.  Tao&#x27;s proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence &quot;backwards&quot;. In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao&#x27;s argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.\n},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Tao has recently proved that if T<sub>1</sub>,&hellip;,T<sub>l</sub> are commuting, invertible, measure-preserving transformations on a dynamical system then for any L<sub>&infin;</sub> functions f<sub>1</sub>,&hellip;f<sub>l</sub>, the average (1/N)&Sigma;<sub>n=0</sub><sup>N-1</sup>&Pi;<sub>i&leq;l</sub>f<sub>i</sub> o T<sup>in</sup> converges in the L<sub>2</sub> norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence \"backwards\". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"towsner-ordinalanalysisbytransformations-2009\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/KPiMu.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-ordinalanalysisbytransformations-2009', 'http://www.sas.upenn.edu/~htowsner/KPiMu.pdf', event);\">\n    Ordinal analysis by transformations.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Ann. Pure Appl. Logic</i>, 157(2-3): 269–280.  2009.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1016/j.apal.2008.09.011\"\n     onclick=\"log_download('towsner-ordinalanalysisbytransformations-2009', 'http://dx.doi.org/10.1016/j.apal.2008.09.011', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Ordinal analysis by transformations [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/KPiMu.pdf\"\n     onclick=\"log_download('towsner-ordinalanalysisbytransformations-2009', 'http://www.sas.upenn.edu/~htowsner/KPiMu.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Ordinal analysis by transformations [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.apal.2008.09.011\"\n     onclick=\"log_download('towsner-ordinalanalysisbytransformations-2009', 'http://doi.org/10.1016/j.apal.2008.09.011', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-ordinalanalysisbytransformations-2009\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>36 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-ordinalanalysisbytransformations-2009')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2499713,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {Ordinal analysis by transformations},\n   JOURNAL = {Ann. Pure Appl. Logic},\n  FJOURNAL = {Annals of Pure and Applied Logic},\n    VOLUME = {157},\n      YEAR = {2009},\n    NUMBER = {2-3},\n     PAGES = {269--280},\n      ISSN = {0168-0072},\n     CODEN = {APALD7},\n   MRCLASS = {03F05 (03F15 03F35)},\n  MRNUMBER = {2499713 (2010e:03069)},\nMRREVIEWER = {Thomas Strahm},\n       DOI = {10.1016/j.apal.2008.09.011},\n       URLJOURNAL= {http://dx.doi.org/10.1016/j.apal.2008.09.011},\nurlpdf={http://www.sas.upenn.edu/~htowsner/KPiMu.pdf},\nabstract={The technique of using infinitary rules in an ordinal analysis has been one of the\nmost productive developments in ordinal analysis. Unfortunately, one of the most\nadvanced variants, the Buchholz &amp;Omega;&lt;sub&gt;&amp;mu;&lt;/sub&gt;-rule, does not apply to systems much stronger\nthan &amp;Pi;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;1&lt;/sub&gt;-comprehension. In this paper, we propose a new extension of the \n rule using game-theoretic quantifiers. We apply this to a system of inductive definitions\nwith the strength of a recursively inaccessible ordinal.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz &Omega;<sub>&mu;</sub>-rule, does not apply to systems much stronger than &Pi;<sup>1</sup><sub>1</sub>-comprehension. In this paper, we propose a new extension of the rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with the strength of a recursively inaccessible ordinal.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2005\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2005')\"\n      onkeypress=\"toggleGroup('group_2005')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2005</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-epsilonsubstitutionfortransfiniteinduction-2005\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/esubTI.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-epsilonsubstitutionfortransfiniteinduction-2005', 'http://www.sas.upenn.edu/~htowsner/esubTI.pdf', event);\">\n    Epsilon substitution for transfinite induction.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Arch. Math. Logic</i>, 44(4): 397–412.  2005.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1007/s00153-004-0241-3\"\n     onclick=\"log_download('towsner-epsilonsubstitutionfortransfiniteinduction-2005', 'http://dx.doi.org/10.1007/s00153-004-0241-3', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Epsilon substitution for transfinite induction [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/esubTI.pdf\"\n     onclick=\"log_download('towsner-epsilonsubstitutionfortransfiniteinduction-2005', 'http://www.sas.upenn.edu/~htowsner/esubTI.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Epsilon substitution for transfinite induction [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-004-0241-3\"\n     onclick=\"log_download('towsner-epsilonsubstitutionfortransfiniteinduction-2005', 'http://doi.org/10.1007/s00153-004-0241-3', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-epsilonsubstitutionfortransfiniteinduction-2005\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>42 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-epsilonsubstitutionfortransfiniteinduction-2005')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2139689,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {Epsilon substitution for transfinite induction},\n   JOURNAL = {Arch. Math. Logic},\n  FJOURNAL = {Archive for Mathematical Logic},\n    VOLUME = {44},\n      YEAR = {2005},\n    NUMBER = {4},\n     PAGES = {397--412},\n      ISSN = {0933-5846},\n     CODEN = {AMLOEH},\n   MRCLASS = {03F05 (03F07 03F30)},\n  MRNUMBER = {2139689 (2006c:03083)},\nMRREVIEWER = {Christian Bennet},\n       DOI = {10.1007/s00153-004-0241-3},\n       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0241-3},\nurlpdf={http://www.sas.upenn.edu/~htowsner/esubTI.pdf},\nabstract = {We apply Mints&#x27; technique for proving the termination of the epsilon\nsubstitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We apply Mints' technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2004\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2004')\"\n      onkeypress=\"toggleGroup('group_2004')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2004</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"towsner-arealizabilityinterpretationforclassicalanalysis-2004\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/realize.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'towsner-arealizabilityinterpretationforclassicalanalysis-2004', 'http://www.sas.upenn.edu/~htowsner/realize.pdf', event);\">\n    A realizability interpretation for classical analysis.\n  </a>\n  \n  \n  \n</span>\n\n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://www.sas.upenn.edu/~htowsner/index.html\">Towsner, H.</a>\n</span>\n\n  \n\n</span>\n\n  <i>Arch. Math. Logic</i>, 43(7): 891–900.  2004.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://dx.doi.org/10.1007/s00153-004-0233-3\"\n     onclick=\"log_download('towsner-arealizabilityinterpretationforclassicalanalysis-2004', 'http://dx.doi.org/10.1007/s00153-004-0233-3', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A realizability interpretation for classical analysis [link]\"\n       title=\"Journal\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Journal</span></a>\n  &nbsp;\n  \n  <a href=\"http://www.sas.upenn.edu/~htowsner/realize.pdf\"\n     onclick=\"log_download('towsner-arealizabilityinterpretationforclassicalanalysis-2004', 'http://www.sas.upenn.edu/~htowsner/realize.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"A realizability interpretation for classical analysis [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00153-004-0233-3\"\n     onclick=\"log_download('towsner-arealizabilityinterpretationforclassicalanalysis-2004', 'http://doi.org/10.1007/s00153-004-0233-3', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/towsner-arealizabilityinterpretationforclassicalanalysis-2004\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>58 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('towsner-arealizabilityinterpretationforclassicalanalysis-2004')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article {MR2096140,\n    AUTHOR = {Towsner, Henry},\n     TITLE = {A realizability interpretation for classical analysis},\n   JOURNAL = {Arch. Math. Logic},\n  FJOURNAL = {Archive for Mathematical Logic},\n    VOLUME = {43},\n      YEAR = {2004},\n    NUMBER = {7},\n     PAGES = {891--900},\n      ISSN = {0933-5846},\n     CODEN = {AMLOEH},\n   MRCLASS = {03F35 (03F05)},\n  MRNUMBER = {2096140 (2005i:03070)},\nMRREVIEWER = {Thomas Strahm},\n       DOI = {10.1007/s00153-004-0233-3},\n       URLJOURNAL= {http://dx.doi.org/10.1007/s00153-004-0233-3},\nurlpdf={http://www.sas.upenn.edu/~htowsner/realize.pdf},\nabstract={We present a realizability interpretation for classical analysis&amp;mdash;an association\nof a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of &amp;Pi;&lt;sub&gt;2&lt;/sub&gt; sentences &amp;forall; x &amp;exist; y A(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).},\n}\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We present a realizability interpretation for classical analysis&mdash;an association of a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of &Pi;<sub>2</sub> sentences &forall; x &exist; y A(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n\n\n\n  </div>\n\n\n  <!-- Modal -->\n\n  <!-- <iframe id=\"bibbase_network_frame\" -->\n  <!--         src=\"//bibbase.org/network/control\" -->\n  <!--         style=\"display: none;\"> -->\n  <!-- </iframe> -->\n\n</div>\n"}; document.write(bibbase_data.data);