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Published online by Cambridge University Press:  19 February 2019

Artem Chernikov
Department of Mathematics, University of California Los Angeles, Los Angeles, CA90095-1555, USA (
Sergei Starchenko
Department of Mathematics, University of Notre Dame, Notre Dame, IN46556, USA (
Margaret E. M. Thomas
Zukunftskolleg, Department of Mathematics and Statistics, University of Konstanz, Box 216, 78457Konstanz, Germany (


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 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 $k-1$.

Research Article
© Cambridge University Press 2019

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