On computing VC-density in VC-minimal theories
In model theory, theories are typically distinguished by the complexity of their definable families. One popular notion of complexity, Vapnik-Chervonenkis density, is borrowed from statistical learning theory. In this talk, I discuss the general notion of computing VC-density in NIP theories, a notion explored by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko in recent work. In this work, they ask if there is a relationship between dp-rank and VC-density. I show a partial result pointing in that direction by studying VC-minimality (a condition stronger than having minimal dp-rank). Any formula in a VC-minimal theory with two parameter variables has VC-density at most two. I conclude by discussing the possibility of extending this result to higher dimensions.