Polylogarithmic bound for the Gyárfás-Sumner conjecture

Abstract

The classical Gyárfás-Sumner conjecture asserts that for any fixed forest $F$, any graph $G$ not containing $F$ as an induced subgraph, with bounded clique number $ω(G)$, has bounded chromatic number $χ(G)$. Recently, Nguyen, Scott, and Seymour proved that every such graph has chromatic number at most $|G|^{o(1)}$. We improve this bound to a polylogarithmic one $(\log |G|)^{O(1)}$.