On large externally definable sets in NIP
Logic
2024-11-20 v3
Abstract
We study cofinal systems of finite subsets of . 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.
Cite
@article{arxiv.2205.11792,
title = {On large externally definable sets in NIP},
author = {Martin Bays and Omer Ben-Neria and Itay Kaplan and Pierre Simon},
journal= {arXiv preprint arXiv:2205.11792},
year = {2024}
}
Comments
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