On uniform definability of types over finite sets

TL;DR

This paper investigates the property of uniform definability of types over finite sets (UDTFS) in model theory, showing its relation to stability, dependence, and dp-minimality, and establishing that all dp-minimal theories possess UDTFS.

Contribution

It introduces and explores UDTFS, demonstrating its presence in stable, weakly o-minimal, and dp-minimal theories, thus advancing understanding of definability properties in model theory.

Findings

Stable and weakly o-minimal theories have UDTFS.

UDTFS implies dependence in theories.

All dp-minimal theories possess UDTFS.

Abstract

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.