Regularity properties of ideals and ultrafilters

For an arbitrary ideal I on the regular cardinal κ we consider the problem of refining a given collection $\text{{}\text{A}{}_{\mathrm{\alpha }}\text{:α < κ}⊆}\text{P}\text{(κ)−I}$ by another collection $\text{{}\text{B}{}_{\mathrm{\alpha }}\text{:α<κ}⊆}\text{P}\text{(k)−I}$ so that the sets in the latter collection are as nearly pairwise disjoint as possible. In this context we discuss regularity of ultrafilters, saturation of ideals and some problems of Fodor and Ulam.