VC-Density in Pairs of Ordered Vector Space

TL;DR

This paper establishes an upper bound on VC-density for formulas in pairs of ordered vector spaces, proves the bound is tight, and shows that dense pairs of o-minimal structures are not dp-minimal, advancing understanding in model theory.

Contribution

It introduces a precise VC-density bound for formulas in pairs of ordered vector spaces and demonstrates its optimality, also linking dense pairs of o-minimal structures to non-dp-minimality.

Findings

VC-density is bounded by twice the number of parameters.

The bound on VC-density is proven to be optimal.

Dense pairs of o-minimal structures are not dp-minimal.

Abstract

We show that the VC-density of any partitioned formula in a pair of ordered vector spaces is bounded above by twice the number of parameter variables. We also show that this bound is optimal and, as a by-product, we prove that no dense pair of o-minimal structures is dp-minimal.