We develop the theory of Cκ, λi, a strongly normal filter over [MATHEMATICAL SCRIPT CAPITAL P]κλ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ [MATHEMATICAL SCRIPT CAPITAL P]κλ: |x | = |x ∩ κ | and |x | is inaccessible} also contains Cκ, λi. We also show that functions can be used to obtain a basis for Cκ, λi.

We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived (...) topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context. (shrink)

We study the distributivity of the bounded ideal on Pkλ and answer negatively to a question of Johnson in [13]. The size of non-normal ideals with the partition property is also studied.