Vaught's Conjecture for Sums of Products of Rooted Trees

Abstract

We confirm Vaught's conjecture for each partial order ${\mathbb X}=\sum _{\mathbb I}\prod _{j 1$ is first-order definable; (f) Each $ω$-categorical partial order from the class $\langle \langle {\mathcal C}^{\rm rt}_{\rm fb}\rangle _Π\rangle _Σ$, where ${\mathcal C}^{\rm rt}_{\rm fb}$ is the class of finite-branching rooted trees, is finitely axiomatizable. Statement (f) is related to the results of Rosenstein (for the class of linear orders) and Schmerl (for the class of partial orders of finite width).