On skinny stationary subsets of

Abstract

AbstractWe introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2λ-saturation of NSκλ ∣ X, where NSκλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NSκλ ∣ X can satisfy neither precipitousness nor 2λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.