Assume Q is a definable subset of a model of T. We define a notion of Q-isolated type, generalizing an earlier definition for countable Q. This notion is absolute. For superstable T, we give some sufficient conditions for the existence of Q-atomic models. We apply this to prove some results on weak categoricity over a predicate.

We give two theories, Th1 and Th2, which are explicitly definable over each other , but are not definitionally equivalent. The languages of the two theories are disjoint.

We try to count the number of countable models M of T with a fixed set Q = φ (M) of realizations of a type φ. Also, for stable T, we define an ordinal rank M measuring multiplicity of types, with additivity properties similar to those of U-rank.

It is shown that no infinite group is interpretable in any structure which is homogeneous in a finite relational language. Related questions are discussed for other ω-categorical structures.