VC-Density in Divisible Oriented Abelian Groups and Their Pairs

TL;DR

This paper establishes bounds on VC-density and dp-rank in theories of divisible oriented abelian groups and their pairs, demonstrating dp-minimality and sharpness of bounds.

Contribution

It provides new bounds on VC-density and dp-rank for these theories, showing dp-minimality and the sharpness of the bounds in pairs.

Findings

VC-density is at most the number of parameters, implying dp-minimality.

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

Pairs of such models have dp-rank exactly 2.

Abstract

We show that the VC-density in certain theories of oriented abelian groups is at most the size of parameter variables, which yields dp-minimality. We further prove that the VC-density of formulas in pairs of such models is bounded by twice the size of parameter variables. This uniform upper bound is shown to be sharp, and as a consequence, we show that such pairs have dp-rank 2.