The clique chromatic number of a graph is the minimum colours needed to colour its vertices so that no inclusion-wise maximal which not an isolated vertex monochromatic. We show every maximum degree $\Delta$ has $O\left(\frac{\Delta}{\log~\Delta}\right)$. obtain as corollary $n$-vertex $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi (G)>\aleph _0$ $2^{\aleph _0}$) then $G$ contains all the finite subgraphs of shift $\mathrm {Sh}_n(\omega )$ for some $n$. a variant this theorem graphs interpretable in stationary stable theories. Furthermore, $\operatorname {U}(G)\leq 2$ we $n\leq suffices.