Thin stationary sets and disjoint club sequences

Abstract

We describe two opposing combinatorial properties related to adding clubs to ω 2 : the existence of a thin stationary subset of P ω1 (ω 2 ) and the existence of a disjoint club sequence on ω 2 . A special Aronszajn tree on ω 2 implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of ω 2 which cannot acquire a club subset by any forcing poset which preserves ω 1 and ω 2 . We prove that the existence of a disjoint club sequence follows from Martin's Maximum and is equiconsistent with a Mahlo cardinal.