OPEN - $500

If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then
\[\mathrm{ex}(n;H) \ll n^{2-1/r}.\]

Conjectured by Erdős and Simonovits. Open even for $r=3$. Alon, Krivelevich, and Sudakov [AKS03] have proved
\[\mathrm{ex}(n;H) \ll n^{2-1/4r}.\]
They also prove the full Erdős-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one component is $r$.