Omega-categorical limits of betweenness relations and $D$-sets
Asma Ibrahim Almazaydeh, Samuel Braunfeld, and Dugald Macpherson

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 -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 -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 -sets grows faster than exponentially. The automorphism group is maximal-closed in the symmetric group. Similar results are shown for the construction in…
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Taxonomy
TopicsAdvanced Algebra and Logic
