Ramsey expansions of $Λ$-ultrametric spaces
Samuel Braunfeld
TL;DR
This work develops Ramsey expansions for the generic finite $\Lambda$-ultrametric space when $\Lambda$ is a finite distributive lattice, using subquotient orders to obtain Ramsey lifts and a concrete description of the universal minimal flow of $\mathrm{Aut}(\Gamma)$. A key technical advance is handling non-unary algebraic closure via a two-tier lift and employing quantifier-free reinterpretations to transfer Ramsey properties to well-equipped lifts, enabling a comprehensive census of homogeneous finite-dimensional permutation structures. The results show that all known such structures arise as well-equipped lifts of $\Lambda$-ultrametric spaces and establish Ramsey properties for Braun’s construction families. Together, these developments provide a unified framework for Ramsey theory in finite languages and connect it to the dynamics of automorphism groups through the universal minimal flow.
Abstract
For a finite lattice $Λ$, $Λ$-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When $Λ$ is finite and distributive, there exists a generic $Λ$-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".