OPEN - $1000

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 $K_3$.

This property holds for $\beta=2$ and not for $3\leq \beta <\omega$ (Specker [Sp57]) and for $\beta=\omega$ (Chang [Ch72]).

The first open case is $\beta=\omega^2$ (see [591]). 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.

See also [590].