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For every orientable surface of finite negative Euler characteristic, we finda right-angled Artin group of cohomological dimension two which does not embedinto the associated mapping class group. For a right-angled Artin group on agraph $\gam$ to embed into the mapping class group of a surface $S$, we showthat the chromatic number of $\gam$ cannot exceed the chromatic number of theclique graph of the curve graph $\mathcal{C}(S)$. Thus, the chromatic number of$\gam$ is a global obstruction to embedding the right-angled Artin group$A(\gam)$ into the mapping class group $\Mod(S)$.

An obstruction to embedding right-angled Artin groups in mapping class groups