Conant's generalised metric spaces are Ramsey

TL;DR

This paper extends the Ramsey property to classes of generalized metric spaces with distances from a linearly ordered monoid, broadening previous results and removing the semi-archimedean restriction.

Contribution

It provides new Ramsey expansions for generalized metric spaces with monoid-based distances, without needing the monoid to be semi-archimedean.

Findings

Established Ramsey properties for classes of generalized metric spaces with ordered monoid distances.

Extended previous results to non-semi-archimedean monoids.

Connected Ramsey properties with extension properties for automorphisms.

Abstract

We give Ramsey expansions of classes of generalised metric spaces where distances come from a linearly ordered commutative monoid. This complements results of Conant about the extension property for partial automorphisms and extends an earlier result of the first and the last author giving the Ramsey property of convexly ordered $S$-metric spaces. Unlike Conant's approach, our analysis does not require the monoid to be semi-archimedean.