Problem Statement
If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then\[\mathrm{ex}(n;H) \ll n^{2-1/r}.\]
Categories:
Graph Theory Turan Number
Progress
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 side of the bipartition is $r$.See also [113] and [147].
See also the entry in the graphs problem collection.
Source: erdosproblems.com/146 | Last verified: January 13, 2026