Erdős #592 · set theory / ramsey theory

Such $\alpha$ (in other words, those such that $\alpha \to (\alpha,3)^2$) are called partition ordinals.

• Specker [Sp57] proved this holds for $\beta=2$ and not for $3\leq \beta <\omega$.
• Chang [Ch72] proved this holds for $\beta=\omega$.
• 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 countable ordinal $\gamma$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property.
• Schipperus [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. 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], and [1169] for the case $\alpha=\omega_1^2$.

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