Ramsey expansions of $S$-metric spaces

Source abstract

Let $S\subseteq \mathbb R^+$ be a subset of positive reals. We say that a metric space is an $S$-metric space if the range of the metric is a subset of $S\cup \{0\}$. In 2007, Delhommé, Laflamme, Pouzet, and Sauer characterized those sets $S$ for which the class of all finite $S$-metric spaces has the amalgamation property. Given an amalgamation class, a natural question is what its Ramsey expansion is. Ramsey expansions are known for classes of $S$-metric spaces for which $S\cup\{0\}$ carries a natural distance monoid structure, for example for $S=\mathbb Q^+$ or when $S$ is topologically closed. In this paper, we extend this result to all amalgamation classes of finite $S$-metric spaces, thereby completing the Ramsey-theoretic classification of amalgamation classes of finite $S$-metric spaces.