Fat sets and saturated ideals

Abstract

We strengthen a theorem of Gitik and Shelah [GS] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that NS_κ ↾ S is saturated then κ ∖ S is fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [BT], showing in particular that if I is a λ⁺⁺⁺-saturated normal ideal on P_κ λ then the conditions of being λ⁺-preserving, weakly presaturated, and presaturated are equivalent for I.