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
$\mathbb{R}$
, 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 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
$k-1$
.
RAMSEY GROWTH IN SOME NIP STRUCTURES