Thin stationary sets and disjoint club sequences

Abstract. We describe two opposing combinatorial properties related to adding clubs to ω2: the existence of a thin stationary subset of Pω1 (ω2) and the exis-tence of a disjoint club sequence on ω2. A special Aronszajn tree on ω2 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 ω2 which cannot acquire a club subset by any forcing poset which preserves ω1 and ω2. We prove that the existence of a disjoint club sequence follows from Martin’s Maximum and is equiconsistent with a Mahlo cardinal. Suppose that S is a fat stationary subset of ω2, that is, for every club set C ⊆ ω2, S ∩ C contains a closed subset with order type ω1 + 1. A number of forcing posets have been defined which add a club subset to S and preserve cardinals under various assumptions. Abraham and Shelah [1] proved that, assuming CH, the poset consisting of closed bounded subsets of S ordered by end-extension adds a club subset to S and is ω1-distributive. S. Friedman [5] discovered a different poset for adding a club subset to a fat set S ⊆ ω2 with finite conditions 1. Thi