Publikation: Ramsey growth in some NIP structures
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2021
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Journal of the Institute of Mathematics of Jussieu. Cambridge University Press. 2021, 20(1), pp. 1-29. ISSN 1474-7480. eISSN 1475-3030. Available under: doi: 10.1017/S1474748019000100
Zusammenfassung
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 .
Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
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o-minimality; NIP; Ramsey's theorem; p-minimality; polynomially bounded
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ISO 690
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. 2021, 20(1), pp. 1-29. ISSN 1474-7480. eISSN 1475-3030. Available under: doi: 10.1017/S1474748019000100BibTex
@article{Chernikov2021Ramse-53535,
year={2021},
doi={10.1017/S1474748019000100},
title={Ramsey growth in some NIP structures},
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.}
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<dcterms:abstract xml:lang="eng">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 .</dcterms:abstract>
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