Problem 592

Erdős Problem #592: declared status 'open'. Formalized: yes. Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then 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: This property holds for $\beta=2$ and not for $3\leq \beta <\omega$ (Specker [Sp57]) and for $\beta=\omega$ (Chang [Ch72]). Galvin and Larson [GaLa74] have shown that if $\beta\geq 3$ has this property then $\beta$ must be 'additively indecomposable', so that in particular $\beta=\omega^\gamma$ for some $\gamma<\omega_1$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property. Prize: $1000. Tags: ramsey theory, set theory.