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 $$c_F$$ c F such that are graphs of arbitrarily large chromatic number and the same clique as in which -free induced subgraph has most . This generalises recent theorems Briański, Davies Walczak, Carbonero, Hompe, Moore Spirkl. Our results imply $$r\geqslant 3$$ r ⩾ 3 class $$K_r$$ K very strong vertex Ramsey-type property, giving vast generalisation result Folkman from 1970. also related tournaments, hypergraphs infinite families graphs, show an analogous statement where replaced by odd girth.