  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 $\gam$ to embed into the mapping class group of a surface $S$, we show
that the chromatic number of $\gam$ cannot exceed the chromatic number of the
clique 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 $\Mod(S)$.
