A Hanf number for saturation and omission

Suppose t = (T; T₁; p) is a triple of two countable theories T ⊈ T₁ in vocabularies ⊤ ⊂ ⊤₁ and ⊤₁-type p over the empty set. We show that the Hanf number for the property 'there is a model M₁ of T₁ which omits p, but M₁⌈⊤ is saturated' is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between 'first order' and 'second order quantification'. In particular, we show that if κ is uncountable, then h³(Lε, ε (Q); κ) = h ³(Lω1 ω1 κ), where h³ is the 'normal' notion of Hanf function (Definition 4.12).