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   2 ω        2    ω          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   2 ω        2    ω          countable models or   6 a 3 b        6    a      3    b          countable models, where  a  and  b  are natural numbers.
['B. Sh. Kulpeshov', 'S. V. Sudoplatov']
Vaught's conjecture for quite o-minimal theories