A short proof of a partition relation for triples

Abstract

We provide a much shorter proof of the following partition theorem of P. Erdős and R. Rado: If $X$ is an uncountable linear order into which neither $\omega_1$ nor $\omega_1^{*}$ embeds, then $X \to (\alpha, 4)^{3}$ for every ordinal $\alpha < \omega + \omega$. We also provide two counterexamples to possible generalizations of this theorem, one of which answers a question of E. C. Milner and K. Prikry.