Conjectured by Erd\H{o}s and Simonovits \cite{ErSi84}. Open even for $r=2$. Alon, Krivelevich, and Sudakov \cite{AKS03} have proved\[\mathrm{ex}(n;H) \ll n^{2-1/4r}.\]They also prove the full Erd\H{o}s-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one side of the bipartition is $r$.

See also [113] and [147].

This problem is #43 in Extremal Graph Theory in the graphs problem collection.


References


[AKS03] Alon, Noga and Krivelevich, Michael and Sudakov, Benny, Tur\'{a}n numbers of bipartite graphs and related
Ramsey-type questions. Combin. Probab. Comput. (2003), 477-494.

[ErSi84] Erd\H{o}s, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (Waterloo, Ont., 1982) (1984), 203-218.