The Borel Complexity of Isomorphism for Theories with Many Types

During the Notre Dame workshop on Vaught’s Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for models of a first order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types. I am grateful to the logic group at Notre Dame for organizing this stimulating workhop.