Let $\alpha$ be a cardinal or ordinal number or an order type such that every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue $K_3$. For every $n\geq 3$ must every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue $K_n$?

Conjectured by Erdős and Hajnal.