Sparse metric hypergraphs

Given a metric space $(X, \rho)$, we say $y$ is between $x$ and $z$ if $\rho(x,z) = \rho(x,y) + \rho(y,z)$. A metric space gives rise to a 3-uniform hypergraph that has as hyperedges those triples $\{ x,y,z \}$ where $y$ is between $x$ and $z$. Such hypergraphs are called metric and understanding them is key to the study of metric spaces. In this paper, we prove that hypergraphs where small subsets of vertices induce few edges are metric. Additionally, we adapt the notion of sparsity with respect to monotone increasing functions, classify hypergraphs that exhibit this version of sparsity and prove that they are metric.Comment: 6 pages, 16 figure

Metric spaces in which many triangles are degenerate

Richmond and Richmond (Amer. Math. Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In fact, $\Theta(n^2)$ suitably placed degenerate triangles suffice

Finite Sholander Trees, Trees, and their Betweenness

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.Comment: 8 page