The Hanf number in the strictly stable case

We associate Hanf numbersH ( t ) to triples t = ( T , T 1 , p ) where T and T 1 are theories and p is a type. We show that the Hanf number for the property: “there is a model M 1 ofT 1which omits p , butM 1 ↾ τ is saturated” is larger than the Hanf number ofL λ + , κbut smaller than the Hanf number ofL( 2 λ ) + , κwhen T is stable with κ = κ ( T ) . In fact, surprisingly, we even characterise the Hanf number oft when we fix ( T , λ ) where T is a first order complete (and stable), λ ≥ | T | and demand| T 1 | ≤ λ .