On the Density of Critical Graphs with No Large Cliques

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $(k - 1)$-colorable, $L$ a list-assignment for $G$, then \textit{$L$-critical} not $L$-colorable but induced is. In 2014, Kostochka Yancey proved lower bound on the average degree an $n$-vertex $k$-critical tending to $k \frac{2}{k 1}$ large $n$ that tight infinitely many values $n$, they asked how their may be improved graphs containing clique. Answering this question, we prove $\varepsilon \leq 2.6\cdot10^{-10}$, $k$ sufficiently $K_{\omega + 1}$-free $L$-critical where $\omega k \log^{10}k$ such $|L(v)| 1$ all $v\in V(G)$, at least $(1 \varepsilon)(k 1) \varepsilon \omega 1$. This result implies some > 0$, satisfying $\omega(G) \mathrm{mad}(G) \log^{10}\mathrm{mad}(G)$ $\omega(G)$ size largest clique in $\mathrm{mad}(G)$ maximum list-chromatic number most $\left\lceil (1 \varepsilon)(\mathrm{mad}(G) \varepsilon\omega(G)\right\rceil$.