Induced subgraphs of induced subgraphs of large chromatic number

Abstract

We prove that, for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$ in which every $F$-free induced subgraph has chromatic number at most $c_F$. This generalises recent theorems of Bria\'{n}ski, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every $r\geq 3$ the class of $K_r$-free graphs has a very strong vertex Ramsey-type property, giving a vast generalisation of a result of Folkman from 1970. We also prove related results for tournaments, hypergraphs and infinite families of graphs, and show an analogous statement for graphs where clique number is replaced by odd girth.