Dependent finitely homogneneous rosy structures

Abstract

We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $\ddot{I}‰$-stable $\ddot{I}‰$-categorical structures. In particular, we prove that such structures have finite þ-rank and are coordinatized by a þ-rank 1 set. We show that they admit a distal, finitely axiomatizable, expansion. These results show that there are, up to inter-definability, at most countably many dependent rosy structures M which are homogeneous in a finite relational language.