On VC-Density in VC-Minimal Theories

Abstract

We show that any formula with two free variables in a Vapnik–Chervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where ${\mathrm{acl}}^{\mathrm{eq}}={\mathrm{dcl}}^{\mathrm{eq}}$, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).