On chromatic number of countable graphs

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.