Ore's asymptotic density conjecture for critical graphs without large cliques

Let $k$ be an integer with $k\geq 4$, and let $G$ be a $k$-critical graph, meaning that $\chi(G)=k$ and every proper subgraph of $G$ has chromatic number less than $k$. Let $K_{k-2}$ denote the complete graph on $k-2$ vertices. Ore's conjecture. For every $k\geq 4$, there exists $\varepsilon_k>0$ such that, if $G$ is $k$-critical and does not contain a $K_{k-2}$ subgraph, then

$$|E(G)|\geq \left(\frac{k}{2}-\frac{1}{k-1}+\varepsilon_k\right)|V(G)|-\frac{k(k-3)}{2(k-1)}.$$

This conjecture asserts a positive asymptotic improvement over the Kostochka–Yancey lower bound for critical graphs excluding large cliques; the general statement remains open, with the paper addressing difficult cases beginning at $k=6$.