Vaught's conjecture for unidimensional theories

In Bue93b] we proved Vaught's conjecture for all superstable theories of nite rank; that is, such a theory has countably many or continuum many countable models. While this proof does settle Vaught's conjecture for unidimensional theories a sharper result can be obtained for these theories and in places the proof can be simpliied. Let T be a properly unidimensional theory with < 2 @ 0 many countable models. First, we prove for T what is called the Tree Theorem. Loosely, it says that a countable model M is almost atomic over a set t(M) M on which there is a partial order such that (t(M);) is a tree and, for a 2 t(M); tp(a=fb 2 t(M) : b ag) has U ? rank 1 and nite multiplicity. This leads to a characterization of the possible structure groups (see Bue93b]). In particular, one set of conditions in which a superstable theory of nite rank can have continuum many models is eliminated for unidimensional theories. 1 Preliminaries Vaught's conjecture says that a countable rst order theory has countably many or 2 @ 0 many countable models. A theory is unidimensional if all nonalgebraic types are nonorthogonal. Vaught's conjecture for superstable theories of nite rank was proved in Bue93b], of which this paper is a sequel. The purpose of this paper is twofold. First, the unidimensional hypothesis leads to sharper results. Speciically, we will give a precise characterization of the possible structure groups and prove that the corresponding class of abelian structures always has nite representation type (see Bue93b, x6]). The author wishes to thank Ludomir Newelski for many helpful conversations on the reuslts in this paper and for nding errors in some earlier proofs.