ABSTRACT. We show that for many pairs of infinite cardinals κ ą µ`ą µ, pκ`, κq ։ pµ`, µq is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that pκ`, κq ։ pµ`, µq for every successor cardinal κ and every µ ă κ, answering a question of Foreman.

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of strongly proper forcing. We prove: (1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman [8] about ideal projections which is more comprehensive and simpler than the solution obtained in [4]. (2) We solve an older question from Foreman [9] about the relationship between generic hugeness and generic almost hugeness. (3) Finally, we provide solutions to two technical questions from Foreman [7] and [8] related to his Duality Theorem.

From large cardinals we show the consistency of normal, fine, κcomplete λ-dense ideals on Pκ(λ) for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.