Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $m$. Does
\[\lim_{n\to \infty}\frac{g_k(n)}{\log n}\]
exist?

It is known that
\[\frac{4}{\log k}\log n\leq g_k(n) \leq \frac{2}{\log(k-2)}\log n+1,\]
the lower bound due to Kostochka [Ko88] and the upper bound to Erdős [Er59b].