It is easy to show that there exists an independent set of size $\gg n^{1/3}$.

In other words, this question asks whether $R(3,3,m) \ll m^{3-c}$ for some $c>0$. This was disproved by Alon and Rödl [AlRo05], who proved that\[\frac{1}{(\log m)^{4+o(1)}}m^3 \ll R(3,3,m) \ll \frac{\log\log m}{(\log m)^2}m^3.\]As reported in [AlRo05] Sudakov has observed that the $\log\log m$ in the upper bound can be removed.

See also [553].

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