PROVED
This has been solved in the affirmative.
Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $\gg \log n$ many vertices. Must $G$ contain $\gg n^{5/2}$ induced subgraphs which pairwise differ in either the number of vertices or the number of edges?
A problem of Erdős, Faudree, and Sós, who proved there exist $\gg n^{3/2}$ many such subgraphs, and note that $n^{5/2}$ would be best possible. (Although in
[Er93] Erdős credits this question to Alon and Bollobás.)
This was proved by Kwan and Sudakov
[KwSu21].
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Additional thanks to: Zach Hunter
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 #636, https://www.erdosproblems.com/636, accessed 2026-04-11
