Successive cardinals with no partial square

We construct a model in which for all 1 ≤ n < ω, there is no stationary subset of אn+1 ∩ cof(אn) which carries a partial square. Let κ be an uncountable cardinal and let μ ≤ κ be regular. A set A ⊆ κ∩cof(μ) is said to carry a partial square if there exists a sequence ⟨cα : α ∈ A⟩ such that each cα is club in α with order type μ, and for all γ which is a common limit point of cα and cβ , cα ∩ γ = cβ ∩ γ. Such a sequence is called a partial square sequence. It was shown by Shelah [6] that if μ < κ are regular cardinals, then there exists a stationary subset of κ ∩ cof(μ) which carries a partial square. In a model of Magidor [4] which satisfies a strong form of stationary set reflection, there is no stationary subset of א2 ∩ cof(א1) which carries a partial square (this was pointed out by several authors; see [7] and [5]). The exact consistency result was obtained in [3], where we showed that the existence of a greatly Mahlo cardinal is equiconsistent with the statement that for some regular uncountable cardinal κ, there is no stationary subset of κ ∩ cof(κ) which carries a partial square. In this paper we demonstrate how to obtain models in which there are successive cardinals with no partial square. Specifically, starting with an increasing sequence ⟨κn : 1 ≤ n < ω⟩ of supercompact cardinals, we collapse each κn to become אn+1 in such a way that in the final model, for all 1 ≤ n < ω there is no stationary subset of אn+1 ∩ cof(אn) which carries a partial square.