On VC-minimal theories and variants

TL;DR

This paper investigates various notions related to VC-minimal theories, introducing new concepts like convex orderability, weak VC-minimality, and full VC-minimality, and explores their relationships and properties.

Contribution

It defines and situates new variants of VC-minimality within the hierarchy of model-theoretic properties, providing insights into their implications and characteristics.

Findings

Convex orderability lies between VC-minimality and dp-minimality.

Weak VC-minimality is between VC-minimality and dependence, with unstable theories interpreting infinite linear orders.

Full VC-minimality is between weak o-minimality and VC-minimality, with theories having low VC-density.

Abstract

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.