Conant's generalised metric spaces are Ramsey

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.
Comment: 25 pages, 4 figures. Corrected proof of Lemma 6.20. Accepted to Contributions to Discrete Mathematics