Classifying ℵ0-categorical theories

Among the complete ℵ0-categorical theories with finite non-logical vocabularies, we distinguish three classes. The classification is obtained by looking at the number of bound variables needed to isolated complete types. In classI theories, all types are isolated by quantifier free formulas; in classII theories, there is a leastm, greater than zero, s.t. all types are isolated by formulas in no more thanm bound variables: and in classIII theories, for eachm there is a type which cannot be isolated inm or fewer bound variables. ClassII theories are further subclassified according to whether or not they can be extended to classI theories by the addition of finitely many new predicates. Alternative characterizations are given in terms of quantifier elimination and homogeneous models. It is shown that for each primep, the theory of infinite Abelian groups all of whose elements are of orderp is classI when formulated in functional constants, and classIII when formulated in relational constants.