Partitioning triples and partially ordered sets

Abstract:

We prove that if $P$ is a partial order and $P \to (\omega )^1_\omega$, then

[(a)] $P \to (\omega + \omega + 1, 4)^3$, and

[(b)] $P \to (\omega + m, n)^3$ for each $m, n < \omega$.

Together these results represent the best progress known to us on the following question of P. Erdős and others. If $P \to (\omega )^1_\omega$, then does $P \to (\alpha , n)^3$ for each $\alpha < \omega _1$ and each $n < \omega$?