On chromatic number of countable graphs

TL;DR

This paper explores the conditions under which countable graphs have finite or infinite chromatic numbers using model theoretic techniques, providing a comprehensive classification and connecting structural properties with coloring complexity.

Contribution

It offers a complete classification of homogeneous graphs by chromatic number and links model theoretic stability with graph coloring properties.

Findings

Instability in Fraïssé limits implies infinite chromatic number.

Hrushovski constructions yield graphs with arbitrarily large finite chromatic numbers.

Stable and o-minimal definable graphs with infinite chromatic number contain arbitrarily large cliques.

Abstract

This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fra\"{i}ss\'{e} limits, we show that instability forces the chromatic number to be infinite, yielding a complete classification of homogeneous graphs with a finite chromatic number. In contrast, Hrushovski construction always produces graphs of finite chromatic number, though the value can be made arbitrarily large. In tame settings -- such as stable graphs of $U$-rank one and graphs definable in o-minimal structures -- an infinite chromatic number necessarily yields arbitrarily large cliques. These results provide a unified framework connecting structural model theoretic properties with chromatic behavior.