On large externally definable sets in NIP

Abstract

We study cofinal systems of finite subsets of $\ddot{I}{‰}_1$. 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.

v2: Corollary 3.11 in v1 had an erroneous proof; it now appears as Theorem 3.8 with a new proof, and part of section 3 has been moved to an appendix v3: Small local improvements