Global square and mutual stationarity at the ℵn

We give the proof of a theorem of Jensen and Zeman on the existence of a global □□ sequence in the Core Model below a measurable cardinal κκ of Mitchell order (oM(κ)oM(κ)) equal to κ++κ++, and use it to prove the following theorem on mutual stationarity at ℵnℵn.

Let ω1ω1 denote the first uncountable cardinal of VV and set Cof(ω1) to be the class of ordinals of cofinality ω1ω1.

Theorem. If every sequence(Sn)n<ω(Sn)n<ωof stationary setsSn⊆Cof(ω1)∩ℵn+2, is mutually stationary, then there is an inner model with infinitely many inaccessibles(κn)n<ω(κn)n<ωso that for everymmthe class of measurablesλλwithoM(λ)≥κmoM(λ)≥κmis, inVV, stationary inκnκnfor alln>mn>m. In particular, there is such a model in which for all sufficiently largem<ωm<ω, the class of measurablesλλwithoM(λ)≥ωmoM(λ)≥ωmis, inVV, stationary belowℵm+2ℵm+2.