Ramsey expansions of $S$-metric spaces

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 t