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

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