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.