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Partition properties on \(\mathcal P_{\kappa}\lambda\). (English) Zbl 1027.03036

Summary: T. K. Menas [J. Symb. Log. 41, 225-234 (1976; Zbl 0331.02045)] showed there exist \(2^{2^{\lambda^{<\kappa}}}\) normal ultrafilters on \({\mathcal P}_\kappa\lambda\) with the partition property if \(\kappa\) is \(2^{\lambda^{<\kappa}}\)-supercompact. We first show that \(\lambda\)-supercompactness of \(\kappa\) implies the existence of a normal ultrafilter on \({\mathcal P}_\kappa \lambda\) with the partition property. We also prove by a similar technique that \(\text{part}^* (\kappa,\lambda)\) holds if \(\kappa\) is \(\lambda\)-ineffable with \(\text{cf}(\lambda) \geq\kappa\). Note that M. Magidor [Proc. Am. Math. Soc. 42, 279-285 (1974; Zbl 0279.02050)] showed \(\kappa\) is \(\lambda\)-ineffable if \(\text{part}^* (\kappa,\lambda)\) holds, and we proved the converse under some additional assumption [J. Math. Soc. Japan 49, 125-143 (1997; Zbl 0888.03034)].

MSC:

03E02 Partition relations
03E55 Large cardinals
03E05 Other combinatorial set theory
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