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All Random Solved Random Open
OPEN
Show that for any rational $\alpha \in (1,2)$ there exists a bipartite graph $G$ such that \[\mathrm{ex}(n;G)\asymp n^{\alpha}.\] Conversely, if $G$ is bipartite then must there exist some rational $\alpha$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}?\]
A problem of Erdős and Simonovits.

See also the entry in the graphs problem collection.