Strong covering without squares

We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let kappa be a cardinal in V. We say that kappa-covering holds between V and W iff for all X in V with X subseteq ON and V models |X|< kappa, there exists Y in W such that X subseteq Y subseteq ON and V models |Y|< kappa. Strong kappa-covering holds between V and W iff for every structure M in V for some countable first-order language whose underlying set is some ordinal lambda, and every X in V with X subseteq lambda and V models |X|< kappa, there is Y in W such that X subseteq Y prec M and V models |Y|< kappa. We prove that if kappa is V-regular, kappa^+_V= kappa^+_W, and we have both kappa-covering and kappa^+-covering between W and V, then strong kappa-covering holds. Next we show that we can drop the assumption of kappa^+-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that kappa^+_W = kappa^+_V and weaken the kappa^+-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

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