Rationality and finite-range conjectures for VC-density

Resolve parts (a) and (b) of Chernikov’s conjecture: prove that the VC-density of every definable set system in an NIP structure is rational, and, for each o-minimal structure and parameter arity n, belongs to a finite subset of [0,n]∩Q.

Background

The paper states Chernikov’s conjecture in three parts. Part (a) asserts rationality of VC-density for definable set systems in arbitrary NIP structures, while part (b) asserts that, in an o-minimal structure, VC-density is restricted to a finite set of rational values depending only on the parameter arity. The paper proves the integer-valued conclusion corresponding to part (c) for semilinear structures and expansions of the ordered additive group, but explicitly identifies parts (a) and (b) as unresolved. The authors explain that resolving these parts would require a deeper understanding of VC-density for semialgebraic set systems and might reduce to suitable incidence bounds.