In other words, is it true that\[\alpha \to (\alpha, 3)^2?\]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]).

This is true and was proved independently by Schipperus [Sc10] and Darby.

See also [118], and [592] for the general case.

View the LaTeX source

This page was last edited 23 January 2026.