Problem 591

Erdős Problem #591: declared status 'proved'. Formalized: yes. Let $\alpha$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$? Current best: For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$. Prize: $250. Tags: ramsey theory, set theory.