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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$\,&lt;$    on the set$Y\setminus F$    such that for each partial automorphism$\varphi $    (i.e., local automorphism, in Fraïssé’s terminology) of the linear order$\langle Y\setminus F, &lt;\rangle $    the mapping$\mathop {\mathrm {id}}\nolimits _F \cup \varphi $    is a partial automorphism of${\mathbb Y}$   . By theorems of Fraïssé and Pouzet, an infinite structure${\mathbb Y}$    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.

VAUGHT’S CONJECTURE FOR ALMOST CHAINABLE THEORIES