A Silver-like Perfect Set Theorem with an Application to Borel Model Theory

Silvers’s perfect set theorem says that every Π1 equivalence relation over a polish space either has countably many equivalence classes or has a perfect set of inequivalent elements. We prove an analog where the equivalence relation is replaced by a Π1 dependence relation, and the perfect set of inequivalent elements is replaced by a perfect independent set. The dependence relation should satisfy a weak form of the exchange property ; and such a dependence relation can be canonically defined on any model of a superstable theory. Thus we can apply the preceding result to such models ; we use it prove the following theorem : every totally Borel model of a superstable theory is saturated as soon as it is ω1-saturated. Where a totally Borel model is any quotient structure M0/E such that • M0 is a structure in a language without equality symbol, the domain of which is the Baire space ω • E is a Borel equivalence relation on M0 which satisfies the axioms of equality with respect to the primitives of M0 • every first order definable relation of (M0, E) is Borel • the language of M0/E is the language of M0 plus equality. Nota-Bene – Classification theory allows to show that many superstable theories have totally Borel ω1-saturated models.