Aside from the well-known characterization of weakly compact cardinals in terms of the usual partition calculus, I've been wondering if there are other characterizations that are variants of the typical monochromatic partition relations we are considering. More precisely:
Do the following partition relations characterize weakly compact cardinals (all cardinals discussed in the following are inaccessible):
- $\begin{pmatrix} \kappa \\ \kappa \end{pmatrix} \to \begin{pmatrix} \kappa \\ \kappa \end{pmatrix}^2_{\omega, <\omega}$, namely, for any $f: \kappa\times \kappa\to \omega$, there exist $A,B\in [\kappa]^\kappa$ such that $|f''A\times B|<\aleph_0$. Notice that this property implies $\neg \square(\kappa)$ by a result of Todorcevic, so in particular $\kappa$ is weakly compact in $L$. I'm also interested in the case where $<\omega$ is replaced by $<3$.
- What about unpolarized partition relations in the square bracket family? $\kappa \to [\kappa]^2_{\omega}$, i.e. for any $f:[\kappa]^2\to \omega$, there exists $A\in [\kappa]^\kappa$ such that $f''A \neq \omega$. Or maybe we can even require $f''A$ to be finite.
