Ramsey growth in some NIP structures

TL;DR

This paper explores bounds in Ramsey's theorem for relations in NIP structures, extending known results to polynomially bounded o-minimal structures and providing new bounds in distal structures.

Contribution

It generalizes a theorem from semialgebraic to polynomially bounded o-minimal structures and establishes bounds in distal structures, including p-adic fields.

Findings

Generalizes Bukh and Matousek's theorem to o-minimal structures

Shows the theorem does not hold in xp

Provides bounds for definable relations in distal 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. Matou\v{s}ek. "Erd\H{o}s-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 $k-1$.