Vaught's conjecture for weakly o-minimal theories of convexity rank 1 (Q1791051)

The main result of this paper states that the number of countable models of a weakly o-minimal theory of convexity rank $1$ in a countable language is equal to a positive integer different from $2$ or $\omega$ or $2^{\omega}$. If it is less than $2^{\omega}$, then the theory is binary (each formula is equivalent to a Boolean combination of formulas with at most two free variables). The tools used include a detailed analysis of sets of realizations of non-isolated types, weak orthogonality and quasirationality of types, $(p,q)$-splitting formulas and the Rudin-Keisler preorder.