## The Journal of Symbolic Logic

### The topological Vaught's conjecture and minimal counterexamples

Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, E-mail: becker@cs.scarolina.edu

Let G be a Polish topological group, let X be a Polish space, let J: G × XX be a Borel-measurable action of G on X, and let AX be a Borel set which is invariant with respect to J, i.e., a Borel set of orbits. The following statement, or various equivalent versions of it, is known as the Topological Vaught's Conjecture.

Let (G, X, J, A) be as above. Either A contains only countably many orbits, or else, A contains perfectly many orbits.

We say that A contains perfectly many orbits if there is a perfect set PA such that no two elements of P are in the same orbit. (Assuming ¬CH, A contains perfectly many orbits iff it contains 2ℵ0 orbits.) The Topological Vaught's Conjecture implies the usual, model theoretic, Vaught's Conjecture for L ω1ω , since the isomorphism classes are the orbits of an action of the group of permutations of ω; we give details in §0. The “Borel” assumption cannot be weakened for either A or J.

Given a Borel-measurable Polish action (G,X,J) and an invariant Borel set BX, we say that B is a minimal counterexample if (G,X,J,B) is a counterexample to the Topological Vaught's Conjecture and for every invariant Borel C ⊂ B, either C or B\C contains only countably many orbits. This paper is concerned with counterexamples to the Topological Vaught's Conjecture (of course, there may not be any), and in particular, with minimal counterexamples. First, there is a theorem on the existence of minimal counterexamples. This theorem was known for the model theoretic case (it is due to Harnik and Makkai), and is here generalized to arbitrary Borel-measurable Polish actions. Second, we study the properties of minimal counterexamples. We give two different necessary and sufficient conditions for a counterexample to be minimal, as well as some consequences of minimality. Some of these results are proved assuming determinacy axioms.This second part seems to be new even in the model theoretic case.