The Hanf number in the strictly stable case

We associate Hanf numbers (Formula presented.) to triples (Formula presented.) where T and T₁ are theories and p is a type. We show that the Hanf number for the property: “there is a model M₁ of (Formula presented.) which omits p, but (Formula presented.) is saturated” is larger than the Hanf number of (Formula presented.) but smaller than the Hanf number of (Formula presented.) when T is stable with (Formula presented.). In fact, surprisingly, we even characterise the Hanf number of (Formula presented.) when we fix (Formula presented.) where T is a first order complete (and stable), (Formula presented.) and demand (Formula presented.).