SOLVED

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.

This was proved by Kwan and Sudakov [KwSu21].