A note on uniform definability of types over finite sets in partial orders of finite width

TL;DR

This paper investigates whether all finite-width partial orders possess the VC1 property, providing a negative answer and insights into the definability of types over finite sets.

Contribution

It demonstrates that not all finite-width partial orders have the VC1 property, addressing a question in model theory about uniform definability of types.

Findings

Finite-width partial orders do not necessarily have the VC1 property.

Provides counterexamples to the conjecture that all such orders have VC1.

Offers related theoretical remarks on VC density and definability.

Abstract

In "VC density in some theories without the independence property" the authors asked whether any partial order of finite width has the VC1 property (i.e. every formula in one variable has UDTFS in one parameter). We give a negative answer and some related remarks.