A model-thoeretic version of Tarski's theorem

TL;DR

This paper introduces a model-theoretic approach to Tarski's theorem, identifying definable bipartite graphs with near-perfect matchings and strengthening the theorem within this definable framework.

Contribution

It presents a novel definable graph class and extends Tarski's theorem to the definable setting in model theory.

Findings

Identified a class of definable bipartite graphs with near-perfect matchings.

Proved a strengthened version of Tarski's theorem for definable structures.

Demonstrated the applicability of model-theoretic methods to classical theorems.

Abstract

Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.