Core models in the presence of Woodin cardinals
arXiv:math/0303089
Abstract
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin cardinals. However, it is still true that if M is an omega-closed iterate of V then K^M is an iterate of K.
6 pages