The comparison theory of hod pairs below AD + + “ The largest Suslin cardinal is a member of the Solovay sequence ” ∗ †

We develop the basic theory of hod mice below the Largest Suslin Axiom (LSA), which says that the largest Suslin cardinal is a member of the Solovay sequence. We also prove comparison theorem for such hod mice in the context of AD+. This is the first paper in planned sequence of three papers that eventually will establish that Mouse Set Conjecture holds provided there is no inner model satisfying LSA. Over the last two decades the Mouse Set Conjecture (MSC) has become one of the central open problems in inner model theory. One of the main reasons for this is that MSC is intimately connected with the inner model program, which is the program for constructing canonical inner models for large cardinals. Moreover, ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †