  We introduce 0^h (0^handgrenade) as a sharp for an inner model with a proper
class of strong cardinals. If 0^h does not exist then any normal iteration tree
is "almost linear." We exploit this fact to prove the existence of the core
model K in the theory "ZFC + 0^h does not exist." (As of today, non-0^h is
thereby the weakest anti large cardinal assumption under which K can be shown
to exist in ZFC. In this sense we improve earlier work of Dodd, Jensen,
Mitchell, and - partially - Steel.) We indicate that our paper provides the
last step for determining the exact consistency strength of a statement in the
Delfino problem list.
