Suppose t = (T, T_1, p) is a triple of two theories in vocabularies τ ? τ_1 with cardinality λ, T ? T_1 and a τ_1-type p over the empty set that is consistent with T_1. We consider the Hanf number for the property "there is a model M_1 of T_1 which omits p, but M_1τ is saturated". In [2], we showed that this Hanf number is essentially equal to the L?wenheim number of second order logic. In this paper, we show that if T is superstable, then the Hanf number is less than _((2~((2λ)+)))~+.
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