Ramsey growth in some NIP structures

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Chernikov_2-dmntgwgo9r550.pdf
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2021
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Chernikov, Artem
Starchenko, Sergei
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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 .

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Fachgebiet (DDC)
510 Mathematik
Schlagwörter
o-minimality; NIP; Ramsey's theorem; p-minimality; polynomially bounded
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ISO 690CHERNIKOV, 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/S1474748019000100
BibTex
@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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