Ramsey growth in some NIP structures

Type of Publication: Journal article
Publication status: Published
URI (citable link): http://nbn-resolving.de/urn:nbn:de:bsz:352-2-dmntgwgo9r550
Author: Chernikov, Artem; Starchenko, Sergei; Thomas, Margaret E. M.
Year of publication: 2021
Published in: Journal of the Institute of Mathematics of Jussieu ; 20 (2021), 1. - pp. 1-29. - Cambridge University Press. - ISSN 1474-7480. - eISSN 1475-3030
ArXiv-ID: arXiv:1609.05951v3
DOI (citable link): https://dx.doi.org/10.1017/S1474748019000100
Summary:
We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal 163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded o -minimal expansions of R , and show that it does not hold in Rexp . This provides a new combinatorial characterization of polynomial boundedness for o -minimal structures. We also prove an analog for relations definable in P -minimal structures, in particular for the field of the p -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society 366(9) (2014), 5043–5065), we show that in distal structures the upper bound for k -ary definable relations is given by the exponential tower of height k−1 .
Subject (DDC): 510 Mathematics
Keywords: o-minimality; NIP; Ramsey's theorem; p-minimality; polynomially bounded
Link to License: In Copyright
Refereed: Unknown
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CHERNIKOV, Artem, Sergei STARCHENKO, Margaret E. M. THOMAS, 2021. Ramsey growth in some NIP structures. In: Journal of the Institute of Mathematics of Jussieu. Cambridge University Press. 20(1), pp. 1-29. ISSN 1474-7480. eISSN 1475-3030. Available under: doi: 10.1017/S1474748019000100

@article{Chernikov2021Ramse-53535, title={Ramsey growth in some NIP structures}, year={2021}, doi={10.1017/S1474748019000100}, number={1}, volume={20}, issn={1474-7480}, journal={Journal of the Institute of Mathematics of Jussieu}, pages={1--29}, author={Chernikov, Artem and Starchenko, Sergei and Thomas, Margaret E. M.} }

Starchenko, Sergei Ramsey growth in some NIP structures Thomas, Margaret E. M. 2021-04-29T11:47:34Z 2021 eng Chernikov, Artem Chernikov, Artem We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal 163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded o -minimal expansions of R , and show that it does not hold in R<sub>exp</sub> . This provides a new combinatorial characterization of polynomial boundedness for o -minimal structures. We also prove an analog for relations definable in P -minimal structures, in particular for the field of the p -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society 366(9) (2014), 5043–5065), we show that in distal structures the upper bound for k -ary definable relations is given by the exponential tower of height k−1 . 2021-04-29T11:47:34Z Thomas, Margaret E. M. terms-of-use Starchenko, Sergei

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