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 $\mathcal{T}$ 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 $\mathcal{T}$ is a complete monomorphic theory having infinite models, then the number of its non-isomorphic countable models, $I(\mathcal{T},\omega )$, is either equal to $1$ or to $\mathfrak{c}$. In addition, $I(\mathcal{T},\omega )=1$ iff some countable model of $\mathcal{T}$ is simply definable by an $\omega$-categorical linear order on its domain.

13 pages