Erdős Problem #920 — Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.

Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$. Is it true that, for $k\geq 4$, $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$?