Abstract

The paper "Partitions of Products" [DiPH] investigated the polarized partition relation $\begin{pmatrix}\omega\\\omega\\\omega\\\vdots\end{pmatrix} \rightarrow \begin{pmatrix}\alpha_1\\\alpha_1\\\alpha_2\\\vdots \end{pmatrix}$ The relation is consistent relative to an inaccessible cardinal if every α i is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite. Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω