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A partition relation for successors of large cardinals. (English) Zbl 1024.03050

Summary: We address partition problems of Erdős and Hajnal by showing that \(\kappa^+ \to (\kappa^2+1,\alpha)\) for all \(\alpha < \kappa^+\), if \(\kappa^{<\kappa}=\kappa\) and \(\kappa\) carries a \(\kappa\)-dense ideal. If \(\kappa\) is measurable we show that \(\kappa^+ \to (\alpha)^2_n\) for \(n<\omega\), \(\alpha<\Omega\) where \(\Omega\) is a very large ordinal less than \(\kappa^+\) that is closed under all primitive recursive ordinal operations.

MSC:

03E55 Large cardinals
03E02 Partition relations
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