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 $M^*$ of $M$ and a formula $\phi(x,y)$ the following are equivalent: (i) $\phi(x,y)$ is NIP on $M$ (in the sense of Definition 2.1). (ii) Whenever $p(x)\in S_\phi(M^*)$ is finitely satisfiable in $M$ then it is Baire 1 definable over $M$ (in sense of Definition 2.5).