Some partition relations for ideals on \(P_{\kappa}\lambda\)
The author develops techniques for proving partition relations for ideals on \(P_{\kappa}\lambda\). The class of seminormal ideals on \(P_{\kappa}\lambda\) is introduced, and it is shown that ideal-theoretic partition relations for this class are related to the distributivity of the quotient algebra with respect to a given ideal. The author extends a proof of Solovay to show that if cf \(\lambda\) \(>\kappa\) and \(\kappa\) is \(\lambda\)-Shelah then \(\lambda^{<\kappa}=\lambda.\) In earlier papers the author introduced an ideal-theoretic analogue, WC, of weak compactness on ideals on uncountable cardinals. Here a similar concept (also called WC) for ideals on \(P_{\kappa}\lambda\) is introduced and shown to be related to the ideal induced by the \(\lambda\)- Shelah property and to mild \(\lambda\)-ineffability. Results of Carr, Kunen, and Baumgartner are extended to show that ineffable and almost ineffable subsets of \(P_{\kappa}\lambda\) may be characterized in terms of regressive partition relations.
scientific article; zbMATH DE number 4212033
scientific article