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 Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every \(r\geqslant 3\) 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.

我们证明，对于每个至少有一条边的图F ，存在一个常数\(c_F\) ，使得存在任意大色数的图和与F相同的团数，其中每个无F的导出子图都有色最多为\(c_F\)。这概括了 Brianski、Davies 和 Walczak、Carbonero、Hompe、Moore 和 Spirkl 的最新定理。我们的结果表明，对于每个\(r\geqslant 3\)的\(K_r\)无图类，具有非常强的顶点 Ramsey 型属性，给出了 1970 年 Folkman 结果的广泛概括。我们还证明锦标赛、超图和无限图族的相关结果，并显示了图的类似陈述，其中团数被奇数周长代替。