Erdős Problem #917 — Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.

Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number). Is it true that $f_k(n) \gg_k n^2?$ Is it true that $f_6(n)\sim n^2/4?$ More generally, is it true that, for $k\geq 6$, $f_k(n) \sim \frac{1}{2}\left(1-\frac{1}{\lfloor k/3\rfloor}\right)n^2?$