Problem Statement
Let $\alpha$ be the infinite ordinal $\omega^\omega$. 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 $K_3$?
Categories:
Set Theory Ramsey Theory
Progress
A problem of Erdős and Rado. For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$ (a question of Erdős for which he offered \$20).This is true, and was proved by Chang [Ch72]. Milner modified Chang's proof to prove that this remains true if we replace $K_3$ by $K_m$ for all finite $m<\omega$ (a shorter proof was found by Larson [La73]).
See also [591] and [592].
Source: erdosproblems.com/590 | Last verified: January 15, 2026