On VC-Density in VC-Minimal Theories

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 acl(eq) = dcl(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).

Automated summary

This article establishes an upper bound for the VC-density of formulas with two free variables in a Vapnik-Chervonenkis (VC) minimal theory. By slightly modifying the argument, a new proof is provided to show that in a VC-minimal theory where acl(eq) = dcl(eq), the VC-density of a formula is at most the number of free variables.