  Suppose t = (T,T_1, p) is a triple of two theories T subset T_1 in
vocabularies tau subset tau_1 (respectively) of cardinality lambda and a
tau_1-type p over the empty set; in the main case here is with T stable. We
show the Hanf number for the property: "there is a model M_1 of T_1 which omits
p, but M_1 restricted to tau is saturated" is larger than the Hanf number of
L_{lambda^+, kappa} but smaller than the Hanf number of L_{(2^lambda)^+, kappa}
when T is stable with kappa = kappa(T). In fact, we characterize the Hanf
number of t when we fix (T, lambda) where T is a first order complete, lambda >
|T| and demand |T_1| < lambda.
