NIP formulas and Baire 1 definability

Abstract

In this short note, using results of Bourgain, Fremlin, and Talagrand \cite{BFT}, we show that for a countable structure $M$, a saturated elementary extension $M^{\ast }$ of $M$ and a formula $\varphi (x,y)$ the following are equivalent: (i) $\varphi (x,y)$ is NIP on $M$ (in the sense of Definition 2.1). (ii) Whenever $p(x)\in S_{\varphi }(M^{\ast })$ is finitely satisfiable in $M$ then it is Baire 1 definable over $M$ (in sense of Definition 2.5).

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