Adding clubs with square

Abstract. We present a technique for destroying stationary subsets of Pκκ+ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ+). In this paper we continue our study of consistency results concerning the set S(κ, κ+) = {a ∈ Pκκ+: o.t.(a) = (a ∩ κ)+} from [5] and [6]. We present a method for destroying the stationarity of certain subsets of Pκκ+, where κ is inaccessible, using partial square sequences. This method of destroying stationary sets has a variety of applications. We prove several results which support the general theme that the structure of S(κ, κ+) can vary greatly depending on the particular model considered. For example, if κ is supercompact and µ < κ is regular, then there exists a generic extension in which S(κ, κ+) is stationary and for almost all a in S(κ, κ+), a∩κ is singular with cofinality µ. On the other hand, it is relatively consistent that for almost all a in S(κ, κ+), a ∩ κ is measurable with Mitchell order (a ∩ κ)++. We also construct a model i