Infinite Stable Graphs With Large Chromatic Number

Abstract

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $\omega$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.