We shall prove that if a model of cardinality κ can be expanded to a model of a sentence ψ of Lλ+,ω by adding a suitable predicate in more than κ ways, then, it has a submodel of power μ which can be expanded to a model of ψ in $> \mu$ ways provided that λ,κ,μ satisfy suitable conditions.

We prove that every relation-universal model of an ω-stable theory is saturated. We also show there is a large class of ω-stable theories for which every resplendent model is homogeneous.