Independent sets and colorings of $K_{t,t,t}$-free graphs

Abstract

Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $\Delta$ has chromatic number $O(\Delta / \log \Delta)$. This was previously known only for almost bipartite graphs, that is, for subgraphs of $K_{1,t,t}$ (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where $F$ is almost bipartite. We prove this conjecture for all $3$-colorable graphs $F$, i.e. subgraphs of $K_{t,t,t}$, representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erd\H{o}s, Koml\'os, and Szemer\'edi from 1981 asserts that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $\Omega(n \log d / d)$. We prove this conjecture in a strong form for all 3-colorable graphs $F$. More precisely, we show that every $n$-vertex $K_{t,t,t}$-free graph of average degree $d$ contains an independent set of size at least $(1 - o(1)) n \log d / d$, matching Shearer's celebrated bound for triangle-free graphs (the case $t = 1$) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the R\"odl nibble method for constructing independent sets with a Tur\'an-type result on $K_{t,t,t}$-free graphs.