离散度量空间：结构、enumeration和$0$ - $1$定律
Discrete metric spaces: structure, enumeration, and $0$-$1$ laws
摘要：
固定一个整数$r\geq 3$。我们考虑$n$点上的度量空间，使得任意两点之间的距离位于$\{1,..., r\}$中。我们的主要结果描述了他们的近似结构大$n$。因此，我们证明了这些度量空间的个数是$\lceil \frac{r+1}{2}\rceil ^{{n\choose 2} + o(n^2)}$。最近，Kozma、Meyerovitch、Peled和Samotij证明了连续情形下的相关结果。当$r$是偶数时，我们的结构刻画更加精确，并且意味着几乎所有这样的度量空间的所有距离至少都是$r/2$。作为一个简单的结果，当$r$是偶数时，我们将上面的误差项从$o(n^2)$改进为$o(1)$，并且还显示了$\mathcal{L}_r$语言中的标记的一阶$0$-$1$定律，该定律由$r$二元关系组成，$[r]$的每个元素对应一个二元关系。特别地，我们证明了几乎必然理论$T$是$\{r/2,..., r\}$中所有带边色的有限简单完全边色图类的Fraïssé极限的理论.我们的工作可以被看作是一个长期的极端组合学的研究的有色设置的延伸，以及除了收集已知的结构，承认逻辑$0$ - $1$法律。
英文摘要：
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.
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