Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph $G$ on $n$ vertices with chromatic number $k$, such that every proper subgraph has chromatic number $<k$, and $G$ can be made bipartite by deleting $m$ edges. Is it true that $f_k(n)\to \infty$ as $n\to \infty$? In particular, is it true that $f_4(n)\gg \log n$?