On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs F, a graph A and a positive integer r, if every graph B∈F has a 2-vertex-connected subgraph which is not a subgraph of A, then there exists an F-free graph which is vertex r-Ramsey with respect to A. We prove that this sufficient condition for the existence of an F-free graph which is vertex r-Ramsey with respect to A is also necessary for large enough number of colours r. We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.