SOLVED
Let $k\geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ in which every odd cycle has length $> m$. Is it true that
\[f_k(n) \asymp n^{\frac{1}{k-2}}?\]
A question of Erdős and Gallai. Gallai
[Ga63] proved that
\[f_4(n) \gg n^{1/2}\]
and Erdős (unpublished) proved $f_4(n) \ll n^{1/2}$.
This was proved for all $k\geq 4$ by Kierstead, Szemerédi, and Trotter [KST84].
