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

Induced subgraphs of induced subgraphs of large chromatic number