Omega-categorical limits of betweenness relations and $D$-sets

TL;DR

This paper investigates the properties of certain oligomorphic permutation groups related to limits of betweenness and D-relations, revealing their non-homogeneity, NIP status, and automorphism group characteristics.

Contribution

It demonstrates that the limit of D-relations is homogenizable and analyzes its model-theoretic and combinatorial properties, resolving open questions from prior work.

Findings

The limit of D-relations is not homogeneous but homogenizable.

The structure is NIP but not monadically NIP.

The automorphism group is maximal-closed in the symmetric group.

Abstract

We explore two constructions of oligomorphic Jordan permutation groups preserving a `limit of betweenness relations' and a `limit of $D$-relations', from \cite{bhattmacph2006jordan} and \cite{almazaydeh2021jordan} respectively. Several issues left open in \cite{almazaydeh2021jordan} are resolved. In particular it is shown that the `limit of $D$-relations' is not homogeneous in the given language, but is `homogenizable', that is, there is a homogeneous structure over a finite relational language with the same universe and the same automorphism group. The structure is NIP, but not monadically NIP, its age is not well-quasi-ordered under embeddability, and the growth rate of the sequence enumerating orbits on $k$-sets grows faster than exponentially. The automorphism group is maximal-closed in the symmetric group. Similar results are shown for the construction in \cite{bhattmacph2006jordan}.