On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy

Let kappa be a cardinal, and let H-kappa be the class of sets of hereditary cardinality less than kappa ; let tau (kappa) > kappa be the height of the smallest transitive admissible set containing every element of {kappa}boolean ORHkappa. We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H-kappa is as long as tau. (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem. Con(ZFC+omega(2)-Pi (1)(1)-Determinacy) double right arrow double right arrowCon(ZFC+V=K+There Exists a long unfoldable cardinal double right arrow double right arrowCon(ZFC+For AllX(X-# exists) + ''For AllD subset of or equal to omega(1) D is universally Baire double left right arrow There Existsris an element ofR(Dis an element ofL(r)))'', and this is set-generically absolute). We isolate a notion of omega-closed cardinal which is weaker than an omega(1)-Erdos cardinal, and show that this bounds the first long unfoldable: Theorem Let kappa be omega -closed. Then there is a long unfoldable lambda