OPEN

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.