### Strong covering without squares

by Shelah. [Sh:580]

Fundamenta Math, 2000

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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