VC density and dp rank

TL;DR

This paper establishes a precise relationship between VC density and dp rank, showing that strong dependence in theories corresponds to finite VC density, thus linking combinatorial and model-theoretic properties.

Contribution

It provides tight bounds connecting VC density and dp rank, demonstrating that strong dependence is characterized by finite VC density.

Findings

VC density is bounded between dpR(n) and dpR(n)+1

Strong dependence is equivalent to finite VC density

Provides a quantitative link between combinatorial and model-theoretic measures

Abstract

We derive that dpR(n) \leq dens(n) \leq dpR(n)+1, where dens(n) is the supremum of the VC density of all formulas in n parameters, and dpR(n) is the maximum depth of an ICT pattern in n variables. Consequently, strong dependence is equivalent to finite VC density.