We show that for many pairs of infinite cardinals κ > μ+ > μ, (κ+, κ) (μ+, μ) is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that (κ+, κ) (μ+, μ) for every successor cardinal κ and every μ < κ, answering a question of Foreman.

Starting from infinitely many supercompact cardinals, we force a model of ZFC where אω2+1 satisfies simultaneously a strong principle of reflection, called ∆-reflection, and a version of the square principle, denoted (אω2+1). Thus we show that אω2+1 can satisfy simultaneously a strong reflection principle and an anti-reflection principle.

We construct a model for the level by level equivalence between strong compactness and supercompactness in which the least supercompact cardinal κ has its strong compactness indestructible under adding arbitrarily many Cohen subsets. There are no restrictions on the large cardinal structure of our model.

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

Assuming that GCH holds and κ is κ+3-supercompact, we construct a generic extension W of V in which κ remains strongly inaccessible and (α+)HOD < α+ for every infinite cardinal α < κ. In particular the rank-initial segment Wκ is a model of ZFC in which (α+)HOD < α+ for every infinite cardinal α.

We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary sets, and answer a question of Foreman and Magidor ([7]) concerning stationary sequences on the first uncountable cardinals, אn, 1 ≤ n < ω.

Under various appropriate hypotheses it is shown that there is only one determinacy model of the form L(R, μ) in which μ is a supercompact measure on Pω1(R). This gives a positive answer to a question asked by W.H. Woodin in 1983.

Starting from a supercompact cardinal κ, we build a model, in which κ is singular string limit, the singular cardinal hypothesis fails at κ and there are no very good scales at κ. Moreover there is a bad scale at κ, and so weak square fails.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality א0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while 2μ + >> μ++.