Metric hypergraphs and metric-line equivalences

In a metric space $M=(X,d)$, we say that $v$ is between $u$ and $w$ if $d(u,w)=d(u,v)+d(v,w)$. Taking all triples $\{u,v,w\}$ such that $v$ is between $u$ and $w$, one can associate a 3-uniform hypergraph with each finite metric space $M$. An effort to solve some basic open questions regarding finite metric spaces has motivated an endeavor to better understand these associated hypergraphs. In answer to a question posed in arXiv:1112.0376, we present an infinite family of hypergraphs that are non-metric, i.e., they don’t arise from any metric space. Another basic structure associated with a metric space is a binary equivalence on the vertex set, where two pairs are in the same class if they induce the same line. An equivalence that comes from some metric space is a metric-line equivalence. We present an infinite family of so called obstacles, that is, binary equivalences that prevent an equivalence from being a metric-line equivalence.