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arXiv · 2410.05832

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

Asma Ibrahim Almazaydeh·Samuel Braunfeld·Dugald Macpherson
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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}.

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Asma Ibrahim Almazaydeh, Samuel Braunfeld, Dugald Macpherson. 2024-10-08. Omega-categorical limits of betweenness relations and $D$-sets. https://arxiv.org/abs/2410.05832

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