A note on sequences witnessing singularity - following Magidor-Sinapova

We address some question raised in Magidor-Sinapova [7] paper. Suppose V ⊆ W , κ is regular in V , but change its cofinality in W . Are there ”nice” witnesses for such change? This is a basic question and a lot of work was done around it (probably the most prominent are Prikry forcing, Jensen and Dodd-Jensen Covering Lemmas, Mitchell Covering Lemmas). Some ZFC results were proved in Dzamonia-Shelah [1] and in [2]. Recently, Magidor and Sinapova [7] studied a supercompact version of it. In this note we address some question raised in this paper. Let us start with the following: Theorem 0.1 Suppose that 1. V ⊆ W . 2. κ is a regular uncountable cardinal in V . 3. μ > κ is a cardinal in V . 4. In V , 2 = μ. 5. (μ) = ∪ n<ω Qn, for some sequence ⟨Qn | n < ω⟩ of elements of Pκ(μ) of V . 6. In W , (μ) ≥ ((2)) . 7. In W , (μ) is a cardinal. Let ⟨Dα | α < (μ) ⟩ ∈ V be a sequence of clubs in Pκ(μ) of V . Then there is an increasing sequence ⟨Pn | n < ω⟩ such that for every α < (μ) , for all but finitely many n < ω, Pn ∈ Dα.