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