Polarized partition relations (Q2747720)

Let \(\kappa\), \(\lambda\), \(\mu\), \(\nu\) be infinite cardinals. Then the polarized partition relation \(\left(\begin{smallmatrix} \kappa\\ \lambda\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \mu\\ \nu\end{smallmatrix}\right)_\rho\) means the following: For every \(f:\kappa\times \lambda\to\rho\) there are \(A\subseteq\kappa\) and \(B\subseteq\lambda\) such that \(|A|= \mu\) and \(|B|=\nu\) and \(f\) is constant on \(A\times B\). If here \(\rho\) is replaced by \(<\rho\) then the meaning is \(\left(\begin{smallmatrix}\kappa\\ \lambda\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \mu\\ \nu\end{smallmatrix}\right)_\tau\) for all \(\tau< \rho\). \(\left(\begin{smallmatrix}\kappa\\ \lambda\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \mu_0 & \nu_0\\ \nu_0 &\nu_1\end{smallmatrix}\right)\) means: For any \(f:\kappa\times \lambda\to 2\) there are \(A\subseteq\kappa\) and \(B\subseteq\lambda\) such that for some \(i< 2\) we have \(|A|= \mu_i\), \(|B|= \nu_i\) and \(f''A\times B= \{i\}\). Finally \(\left(\begin{smallmatrix} \kappa\\ \lambda\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \mu_0\\ \nu_0\end{smallmatrix} \left(\begin{smallmatrix}\mu_1\\ \nu_1\end{smallmatrix}\right)_\rho\right)\) means: For any \(f:\kappa\times \lambda\to 1+\rho\) there are \(A\subseteq\kappa\) and \(B\subseteq \lambda\) such that either \(|A|= \mu_0\) and \(|B|= \nu_0\) and \(f''A\times B= \{0\}\) or else there is \(i> 0\) such that \(|A|= \mu_1\), \(|B|= \nu_1\), and \(f''A\times B= \{i\}\).NEWLINENEWLINENEWLINEThe main result is now: NEWLINE\[NEWLINE\left(\begin{matrix} (2^{<\kappa})^{++}\\ (2^{<\kappa})^+\end{matrix}\right)\to \left(\begin{matrix} 2^{<\kappa}\\ (2^{<\kappa})^+\end{matrix} \left(\begin{matrix} \kappa\\ \kappa\end{matrix}\right)_{< \text{cf }\kappa}\right).NEWLINE\]NEWLINE If \(\kappa\) is regular this result is optimal.NEWLINENEWLINENEWLINEAnother main theorem states: If \(\kappa\) is a weakly compact cardinal then \(\left(\begin{smallmatrix} \kappa^+\\ \kappa\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \kappa\\ \kappa\end{smallmatrix}\right)_{< \kappa}\) and also \(\left(\begin{smallmatrix} \kappa^+\\ \kappa\end{smallmatrix}\right)\to \left(\begin{smallmatrix} \kappa+1\\ \kappa\end{smallmatrix}\right)_{<\kappa}\). This theorem generalizes former results of \textit{G. V. Choodnovsky} [Infinite and finite sets, Colloq. Hon. P. Erdős, Keszthely 1973, Colloq. Math. Soc. János Bolyai 10, 289-306 (1975; Zbl 0324.02066)], \textit{K. Wolfsdorf} [Arch. Math. Logik Grundlagenforsch. 20, 161-171 (1980; Zbl 0471.03046)] and \textit{A. Kanamori} [Logic colloquium '80, Eur. Summer Meet., Prague 1980, Stud. Logic Found. Math. 108, 153-172 (1982; Zbl 0495.03033)].