A definable $(p,q)$-theorem for NIP theories

TL;DR

This paper establishes a definable $(p,q)$-theorem within NIP theories, extending previous results and answering an open question, with a focus on uniform versions and the concept of locally compressible types.

Contribution

It introduces a definable $(p,q)$-theorem for NIP theories and develops the notion of locally compressible types to generalize prior distal case results.

Findings

Proves a definable $(p,q)$-theorem in NIP theories.

Answers an open question by Chernikov and Simon.

Develops the concept of locally compressible types.

Abstract

We prove a definable version of Matou\v{s}ek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in the distal case, utilizing the notion of locally compressible types which appeared in the work of the author with Bays and Simon.