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All Random Solved Random Open
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There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$
The current bounds are \[ \frac{n^{3/2}}{(\log n)^{3/2}}\ll R(C_4,K_n)\ll \frac{n^2}{(\log n)^2}.\] The upper bound is due to Szemerédi (mentioned in [EFRS78]), and the lower bound is due to Spencer [Sp77].

See also the entry in the graphs problem collection.