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 the unknown `erdos_596.answer`. See Problem 595 for the specific case \((G_1, G_2) = (K_4, K_3)\).