The ACO Seminar (2015–2016)

Fix an integer r ≥ 3. Given an integer n, we define M_r(n) to be the set of metric spaces with underlying set {1,..., n} such that the distance between any two points lies in {1,...,r}. We present results describing the approximate structure of these metric spaces when n is large. We then present consequences of these structural results, including an asymptotic enumeration for M_r(n), and in the case when r is even, a first-order labeled 0-1 law. This is joint work with Dhruv Mubayi.