Baumgartner, James E.; Hajnal, Andras Polarized partition relations. (English) Zbl 0994.03036 J. Symb. Log. 66, No. 2, 811-821 (2001). 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 \(f''A\times B= \{i\}\).The main result is now: \[ \left(\begin{matrix} (2^{<\kappa})^{++}\\ (2^{<\kappa})^+\end{matrix}\right)\to \left(\begin{matrix} 2^{<\kappa}\\ (2^{<\kappa})^+\end{matrix} \left(\begin{matrix} \kappa\\ \kappa\end{matrix}\right)_{< \text{cf }\kappa}\right). \] If \(\kappa\) is regular this result is optimal.Another 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 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)], K. Wolfsdorf [Arch. Math. Logik Grundlagenforsch. 20, 161-171 (1980; Zbl 0471.03046)] and A. Kanamori [Logic colloquium ’80, Eur. Summer Meet., Prague 1980, Stud. Logic Found. Math. 108, 153-172 (1982; Zbl 0495.03033)]. Reviewer: Egbert Harzheim (Köln) Cited in 2 Documents MSC: 03E02 Partition relations Keywords:infinite cardinals; polarized partition relation Citations:Zbl 0324.02066; Zbl 0471.03046; Zbl 0495.03033 PDFBibTeX XMLCite \textit{J. E. Baumgartner} and \textit{A. Hajnal}, J. Symb. Log. 66, No. 2, 811--821 (2001; Zbl 0994.03036) Full Text: DOI References: [1] Proceedings of the colloquium on infinite and finite sets pp 289– (1973) [2] DOI: 10.1016/0168-0072(87)90077-7 · Zbl 0643.03038 [3] Finite and infinite sets (Proceedings of the Banff conference) (1991) [4] Proceedings of the colloquium on infinite and finite sets pp 109– (1973) [5] DOI: 10.1007/BF02021135 · Zbl 0471.03046 [6] Proceedings of the Tarski symposium (Proceedings of the symposium on pure mathematics) pp 269– (1971) [7] Proceedings of the logic colloquium ’80 pp 153– (1982) [8] DOI: 10.1016/0012-365X(72)90060-X · Zbl 0239.04005 [9] Fundamenta Mathematicae 69 pp 39– (1970) [10] DOI: 10.1090/S0002-9904-1956-10036-0 · Zbl 0071.05105 [11] Acta Mathematika Hungarica 16 pp 93– (1965) [12] Fundamenta Mathematicae 155 pp 153– (1998) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.