RAMSEY GROWTH IN SOME NIP STRUCTURES

Abstract

AbstractWe 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 Journal163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it does not 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 Conlon et al. (Transactions of the American Mathematical Society366(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 $k-1$.