# Miloš Kurilić: Vaught’s Conjecture for Monomorphic Theories

Place: Fields Institute (Room 210)
Date: November 30, 2018 (1:00-2:00)
Speaker: Miloš Kurilić
Title: 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.

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