General chromatic cut conjecture

Let $k$ be an integer, let $G$ be a graph with at least $k$ vertices, and let $X$ be one side of a cut of $G$. Write $V(G)$ and $E(G)$ for the vertex and edge sets, and let $\chi(G[X])$ denote the chromatic number of the subgraph induced by $X$. General chromatic cut conjecture. For every integer $k$ and every graph $G$ on at least $k$ vertices, if

$$|E(G)| < k|V(G)| - \frac{k(k+1)}{2},$$

then $G$ admits a cut $X$ with $\chi(G[X])<k$. In particular, $\ell_k=k$. This conjecture generalizes the small cases, including the bipartite-cut assertion for $k=3$, and remains open; the paper records only partial bounds for forest and bipartite cuts.