Dependent finitely homogneneous rosy structures

TL;DR

This paper investigates finitely homogeneous dependent rosy structures, establishing their finite { h}-rank, coordinatization by a { h}-rank 1 set, and the existence of a distal, finitely axiomatizable expansion, revealing their limited diversity.

Contribution

It extends known results from $ ext{omega}$-stable $ ext{omega}$-categorical structures to dependent rosy structures, showing their finite { h}-rank and structural properties.

Findings

Structures have finite { h}-rank.

They are coordinatized by a { h}-rank 1 set.

Existence of a distal, finitely axiomatizable expansion.

Abstract

We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $\omega$-stable $\omega$-categorical structures. In particular, we prove that such structures have finite {\th}-rank and are coordinatized by a {\th}-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.