Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$ vertices has an independent set of size at least $\log n$ must contain an independent set of size at least $f(n)$.Estimate $f(n)$. In particular, is it true that $f((\log n{)}^2,n)\ge n^{1/2-o(1)}$? Is it true that $f((\log n{)}^3,n)\gg (\log n{)}^3$?