OPEN
This is open, and cannot be resolved with a finite computation.
- $500
If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then\[\mathrm{ex}(n;H) \ll n^{2-1/r}.\]
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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].
This problem is
#43 in Extremal Graph Theory in the graphs problem collection.
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This page was last edited 18 January 2026. View history
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #146, https://www.erdosproblems.com/146, accessed 2026-04-11
