Vaught's conjecture for monomorphic theories

Abstract

A complete first order theory of a relational signature is called monomorphic iff all its models are monomorphic (i.e. have all the n-element substructures isomorphic, for each positive integer n). We show that a complete theory T having infinite models is monomorphic iff it has a countable monomorphic model and confirm the Vaught conjecture for monomorphic theories. More precisely, we prove that if T is a complete monomorphic theory having infinite models, then the number of its non-isomorphic countable models, I(T,ω), is either equal to 1 or to c. In addition, the equality I(T,ω)=1 holds iff some countable model of T is simply definable by (or, equivalently, freely interpretable in) an ω-categorical linear order on its domain.