Such $\alpha$ are called partition ordinals.
{UL}
{LI}Specker \cite{Sp57} proved this holds for $\beta=2$ and not for $3\leq \beta <\omega$.{/LI}
{LI} Chang \cite{Ch72} proved this holds for $\beta=\omega$.{/LI}
{LI} Galvin and Larson \cite{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 countable ordinal $\gamma$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property.{/LI}
{LI}Schipperus \cite{Sc10} have proved this is holds if $\beta=\omega^\gamma$ in which $\gamma$ is a countable ordinal which is the sum of one or two indecomposable ordinals, and this fails to hold if $\gamma$ is the sum of four or more indecomposable ordinals.{/LI}
{/UL}
The remaining open case appears to be when $\gamma$ is the sum of three indecomposable ordinals.

The case $\beta=\omega$ is the subject of [590], and $\beta=\omega^2$ is the subject of [591]. See also [118].


References


[Ch72] Chang, C. C., A partition theorem for the complete graph on {$\omega\sp{\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.

[GaLa74] Galvin, Fred and Larson, Jean, Pinning countable ordinals. Fund. Math. (1974/75), 357-361.

[Sc10] Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic (2010), 1195-1215.

[Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.