Erdős Problem 596

Erdős Problem 596* (Erdős–Hajnal, [Er87]). For which graph pairs $(G_1, G_2)$ is it true that

(1) for every $n \geq 1$ there is a graph $H$ without a $G_1$ such that any $n$-colouring of $H$'s edges contains a monochromatic $G_2$, and yet (2) for every graph $H$ without a $G_1$ there is an $\aleph_0$-colouring of $H$'s edges with no monochromatic $G_2$?

Erdős and Hajnal originally conjectured that no such pair exists; but $(C_4, C_6)$ witnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the class of all such pairs, recorded here as answer(sorry).

See Problem 595 for the specific case $(G_1, G_2) = (K_4, K_3)$.