Vaught's conjecture for quite o-minimal theories

Abstract

We study Vaught's problem for quite o-minimal theories. Quite o-minimal theories form a subclass of the class of weakly o-minimal theories preserving a series of properties of o-minimal theories. We investigate quite o-minimal theories having fewer than 2ω countable models and prove that the Exchange Principle for algebraic closure holds in any model of such a theory and also we prove binarity of these theories. The main result of the paper is that any quite o-minimal theory has either 2ω countable models or 6a3b countable models, where a and b are natural numbers.