Ramsey growth in some NIP structures

Abstract

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 [B. Bukh, J. Matoušek. "Erdős-Szekeres-type statements: Ramsey function and decidability in dimension $1$", Duke Mathematical Journal 163.12 (2014): 2243-2270] from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it doesn't hold in ${\mathbb{R}}_{\exp }$. 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 [D. Conlon, J. Fox, J. Pach, B. Sudakov, A. Suk "Ramsey-type results for semi-algebraic relations", 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 $k-1$.

v.3 26 pages; Section 5 was expanded, providing a discussion of polynomial boundedness in this setting and generalizing the proof to demonstrate that the result applies to P-minimal expansions of fields including analytic expansions and finite extensions of Q_p; minor corrections and presentation improvements were made throughout the article

References in corpus (2)

• Distal and non-distal NIP theories
• Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1

Cited by in corpus (1)

• Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications