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Creators/Authors contains: "Thomas, Margaret E."

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  1. ; ; (, Journal of the Institute of Mathematics of Jussieu)
    null (Ed.)
    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 ( 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 $k-1$ . 
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