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
[ErSi84]. Open even for $r=2$. 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 $r$.
See also [113] and [147].
See also the entry in the graphs problem collection.
