A polarized partition relation for cardinals of countable cofinality

Abstract:

We prove that if $\operatorname {cf}{\kappa } = \omega$ and $\lambda = 2^{<\kappa }$, then \[ \left ( \begin {matrix} \lambda ^+ \lambda \end {matrix} \right ) \to \left ( \begin {matrix} \lambda ^+ & \alpha \lambda & \kappa \end {matrix} \right )^{\!\!\!1,1} \] for all $\alpha < \omega _1$. This polarized partition relation holds if for every partition $\lambda \times \lambda ^+ = K_0 \cup K_1$ either there are $B_0 \in [\lambda ]^{\lambda }$ and $A_0 \in [\lambda ^+]^{\lambda ^+}$ with $B_0 \times A_0 \subseteq K_0$ or there are $B_1 \in [\kappa ]^{\lambda }$ and $A_1 \in [\alpha ]^{\lambda ^+}$ with $B_1 \times A_1 \subseteq K_1$.