Infinite Stable Graphs With Large Chromatic Number II

Abstract

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than ${\beth }_2({\aleph }_0)$ then $G$ contains all finite subgraphs of Sh${}_n(\omega )$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.