Vaught's Conjecture for Almost Chainable Theories

Abstract

A structure ${\mathbb Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi$ (i.e., local automorphism, in Fra\"{\i}ss\'{e}'s terminology) of the linear order $\langle Y\setminus F, < \rangle$ the mapping ${\mathrm{id}} _F \cup \varphi$ is a partial automorphism of ${\mathbb Y}$. By a theorem of Fra\"{\i}ss\'{e}, if $|L|<\omega$, then ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer $m$ such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size $n$, for each positive integer $n$. A complete first order $L$-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.