# A consequence of a cardinal being weakly compact

Given $\kappa$ weakly compact, $f: \kappa^+\times \kappa \to \lambda$ where $\lambda<\kappa$ and let $N\prec H(\kappa^{++})$ be such that $|N|=\kappa$ and $\alpha=N\cap \kappa^+\in \kappa^+$ and $\langle N_\xi: \xi<\kappa\rangle$ be an increasing continuous elementary chain such that $|N_\xi|<\kappa$ and $\bigcup_{\xi<\kappa} N_\xi = N$ and there exists a stationary set $T\subset \kappa \backslash \lambda+1$ such that for all $\xi\in T$ $N_\xi\cap \kappa=\xi$, $\xi$ is regular, $N_\xi^{<\xi}\subset N_\xi$.

Assume there exist $\kappa$ many $\xi\in T$ such that for some $A_\xi \subset N_\xi \cap \kappa^+$ and $B_\xi\subset \xi$ with $type(A_\xi)=type(B_\xi)=\xi$, the set $A_\xi\times B_\xi$ is homogeneous for $f$. Why is it true that there exist $A\subset \kappa^+$ and $B\subset \kappa$ with $type(A)=\kappa+1$ and $type(B)=\kappa$ such that $A\times B$ is homogeneous for $f$? It was suggested that $\Pi_1^1$-indescribability would be useful but I did not see precisely how this can be done.

This was a step in the Baumgartner-Hajnal proof of a polarized partition relation for weakly compact cardinals, namely given $f: \kappa^+\times \kappa\to \lambda$ for any $\lambda<\kappa$, there exist $A\in [\kappa^+]^{\kappa^+}, B\in [\kappa]^\kappa$ such that $f''A\times B$ is constant.