In [1] it is proved that the the core model K exists under the assumption that there is no inner model having a Woodin cardinal and satisfying ZFC. Furthermore, they also mention that the result is the "best possible", in the sense that we cannot weaken this anti-large-cardinal hypothesis without strengthening the remainder of the hypothesis.

From what I understand, using the core model induction (which I must admit I don't know the details of yet), K can be constructed below any finite number of Woodin cardinals. So my question is then whether or not K has been constructed below $\omega$ many Woodins and if so, what the extra hypothesis is? If this is indeed the case, I would greatly appreciate a reference to the given paper.

[1] Jensen and Steel: "K without the measurable"