SOLVED
Is there a constant $\delta>0$ such that, for all large $n$, if $G$ is a graph on $n$ vertices which is not Ramsey for $K_3$ (i.e. there exists a 2-colouring of the edges of $G$ with no monochromatic triangle) then $G$ contains an independent set of size $\gg n^{1/3+\delta}$?
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].
