On the density of critical graphs with no large cliques

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