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Discrete metric spaces: structure, enumeration, and $0$-$1$ laws

Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that the distance between any two points lies in $\{1,..., r\}$. Our main result describes their approximate structure for large $n$. As a consequence, we show that the number of these metric spaces is $\lceil \frac{r+1}{2}\rceil ^{{n\choose 2} + o(n^2)}$. Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij. When $r$ is even, our structural characterization is more precise, and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when $r$ is even we improve the error term above from $o(n^2)$ to $o(1)$, and also show a labeled first-order $0$-$1$ law in the language $\mathcal{L}_r$, consisting of $r$ binary relations, one for each element of $[r]$. In particular, we show the almost sure theory $T$ is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\{r/2,..., r\}$. Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical $0$-$1$ laws.