Coloring graphs with forbidden bipartite subgraphs

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph F , there is a constant cF > 0 such that if G is an F -free graph of maximum degree ∆, then χ(G) 6 cF∆/ log ∆. 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 Kt,t-free, then χ(G) 6 (t+o(1))∆/ log ∆ as ∆ → ∞. We improve this bound to (1+o(1))∆/ log ∆, 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.