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|a Fix an integer r ≥ 3. We consider metric spaces on n points such that the distance between any two points lies in. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is (Equation Presented) Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least. As an easy consequence, when r is even, we improve the error term above from to o(n2) to o(1), and also show a labeled first-order 0-1 law in the language Ⅎr, 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. Copyright © 2019 The Association for Symbolic Logic.
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