A finite-color partition relation for $\omega_1^2$ under $\mathrm{MA}_{\aleph_1}$

Abstract

We prove that, assuming $\mathrm{MA}_{\aleph_1}$, for every finite $k\ge 1$,\[\omega_1^2 \to (\omega_1\omega, \underbrace{3,\ldots,3}_{k})^2_{k+1}.\]This gives a conditional affirmative answer to Erd\H{o}s Problem \#1171.The main ingredient is a general color-reduction lemma: if an ordinal $\alpha$ satisfies\[\alpha\to(\alpha,3)^2,\]then for every finite $k\ge1$,\[\alpha\to(\alpha,\underbrace{3,\ldots,3}_{k})^2_{k+1}.\]Applying this lemma to Baumgartner's partition relation\[\omega_1\omega\to(\omega_1\omega,3)^2\]under $\mathrm{MA}_{\aleph_1}$, and using that $\omega_1\omega$ is an initial segment of $\omega_1^2$, yields the result.