OPEN
This is open, and cannot be resolved with a finite computation.
- $250
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 $K_3$?
For comparison, Specker
[Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$. Chang proved this property holds when $\alpha=\omega^\omega$ (see
[590]).
See
[592] for the general case.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #591, https://www.erdosproblems.com/591, accessed 2026-01-14
