Problem #573: Is it true that\[\mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}?

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Progress

A problem of Erdős and Simonovits, who proved that\[\mathrm{ex}(n;\{C_4,C_5\})=(n/2)^{3/2}+O(n).\]Kövári, Sós, and Turán [KST54] proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\sim (n/2)^{3/2}$. This problem is therefore asking whether the threshold is the same if we just forbid odd cycles of length $3$.

See also [574] for the general case, and [765] for $\mathrm{ex}(n;C_4)$.

See also the entry in the graphs problem collection.