On large externally definable sets in NIP

TL;DR

This paper proves that in NIP theories, uncountable externally definable sets always contain infinite definable subsets, extending understanding of definability and structure in model theory.

Contribution

It demonstrates that uncountable externally definable sets in NIP theories always include infinite definable subsets, answering a question posed by Chernikov and Simon in 2013.

Findings

Externally definable sets in NIP theories contain infinite definable subsets.

Cofinal systems of finite subsets of ω₁ can be NIP but not definable in an NIP structure.

Positive results extend to larger cardinals.

Abstract

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.