Ramsey expansions of $\Lambda$-ultrametric spaces

TL;DR

This paper identifies Ramsey expansions for finite distributive lattice-based ultrametric spaces, enabling analysis of their automorphism groups and connecting to homogeneous permutation structures.

Contribution

It introduces new Ramsey expansions for generic finite distributive $ ext{Lambda}$-ultrametric spaces and relates them to the structure of homogeneous permutation structures.

Findings

Established Ramsey properties for these spaces.

Described the universal minimal flow of their automorphism groups.

Connected to the classification of finite-dimensional permutation structures.

Abstract

For a finite lattice $\Lambda$, $\Lambda$-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When $\Lambda$ is finite and distributive, there exists a generic $\Lambda$-ultrametric space, and we here identify a family of Ramsey expansions for that space. This then allows a description the universal minimal flow of its automorphism group, and also implies the Ramsey property for all known homogeneous finite-dimensional permutation structures, i.e. structures in a language of finitely many linear orders. A point of technical interest is that our proof involves classes with non-unary algebraic closure operations. As a byproduct of some of the concepts developed, we also arrive at a natural description of the known homogeneous finite-dimensional permutation structures, completing our previously begun "census".