Coloring graphs with forbidden bipartite subgraphs

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F>0$ such that if $G$ is an $F$-free graph of maximum degree $\mathrm{\Delta }$, then $\chi (G)\le c_F\mathrm{\Delta }\mathrm{/}\log \mathrm{\Delta }$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $\chi(G) \leq (t + o(1)) \Delta / \log\Delta$as$\Delta \to \infty$. We improve this bound to $(1+o(1))\mathrm{\Delta }\mathrm{/}\log \mathrm{\Delta }$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvo\v{r}'ak and Postle.