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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 $Δ$, then $χ(G) \leq c_F Δ/ \logΔ$. 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 $χ(G) \leq (t + o(1)) Δ/ \logΔ$ as $Δ\to \infty$. We improve this bound to $(1+o(1)) Δ/\log Δ$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvořák and Postle.