A polarized partition relation for cardinals of countable cofinality

We prove that if $\operatorname{cf}{\kappa} = \omega$ and $\lambda = 2^{<\kappa}$, then

$$\left( \begin{matrix} \lambda^+ \lambda \end{matrix}\right) \t... ...egin{matrix} \lambda^+ & \alpha \lambda & \kappa \end{matrix}\right)^{1,1}$$

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 $B_1 \times A_1 \subseteq K_1$.