Vaught’s Conjecture for Almost Chainable Theories

Abstract

A structure of a relational language L is called almost chainable iff there are a finite set and a linear order on the set such that for each partial automorphism (i.e., local automorphism, in Fraïssé’s terminology) of the linear order the mapping is a partial automorphism of . By theorems of Fraïssé and Pouzet, an infinite structure is almost chainable iff the profile of is bounded; namely, iff there is a positive integer m such that has non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory having infinite models is called almost chainable iff all models of are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of . 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.