Title: VCd in orders of finite width
Abstract:
In “Vapnik-Chervonenkis density in some theories without the independence property”, the authors discuss some combinatorical and model-theoretic properties related to the VC dimension (probably best known to model theorists from the definition of NIP). Among these properties they discuss properties known as VCd (when d is some natural number) and vc^T(1)=a(for a in R). In particular, they show that weakly quasi-o-minimal structures have the property VC1, which implies they also satisfy vc^T(1)=1. They also show that posets of finite width (that is, posets with a finite bound on the size of antichains) are reducts of a weakly quasi-o-minimal structure, implying that they satisfy vc^T(1)=1 but not necessarily VC1.
This naturally raises the question of whether such orders of finite width are in fact VC1 (a question which the paper leaves open). In this talk, after briefly explaining these properies, we answer the question negatively, and discuss additional related results (and further unsolved problems).