The topic of this talk is inspired by measure-theoretic questions raised by Ulam: What is the smallest number of countably additive, two valued measures on R such that every subset is measurable in one of them?  Under CH, the minimal answer to this question has several equivalent formulations, one of which is the maximal saturation property for ideals on aleph_1, aleph_1-density.  Our goal is to show that these equivalences are special to aleph_1.  In the second talk, we will continue with construction of normal ideals of minimal possible density on a variety of spaces from almost-huge cardinals.  This generalizes a result of Woodin.