Title: The list chromatic number of graphs with small clique number,
Abstract: We prove that every triangle-free graph with maximum degree $\Delta$ has list
chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the
best-known bound for graphs of girth at least 5. We also provide a new proof
that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most
$200r\frac{\Delta\ln\ln\Delta}{\ln\Delta}$.