Right-angled Artin groups and finite subgraphs of curve graphs

We show that for a sufficiently simple surface $S$, a right-angled Artin group $A(\Gamma)$ embeds into $\Mod(S)$ if and only if $\Gamma$ embeds into the curve graph $\mC(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(\Gamma)\to\Mod(S)$ for some $\Gamma$ not contained in $\mC(S)$.