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.