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

Abstract

We explore two constructions of oligomorphic Jordan permutation groups preserving a ‘limit of betweenness relations’ and a ‘limit of $D$-relations’, from Bhattacharjee and Macpherson (2006) and Almazaydeh and Macpherson (2021), respectively. Several issues left open in Almazaydeh and Macpherson (2021) are resolved. In particular, we show 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 and indeed distal, 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 Bhattacharjee and Macpherson (2006).