Metric hypergraphs and metric-line equivalences

Abstract

In a metric space with distance function dist, we say that v is between u and w if u,v,w are pairwise distinct and dist(u,w)=dist(u,v)+dist(v,w). All triples {u,v,w} such that v is between u and w constitute a hypergraph induced by this metric space. In answer to a question posed in [3], we construct an infinite family of 3-uniform hypergraphs induced by no metric space.The line determined by a pair of distinct points x and y in a metric space is the set consisting of x, y, and all points z such that one of x,y,z is between the remaining two. Two pairs of distinct points are said to be equivalent if they determine the same line. The resulting equivalence relation is a metric-line equivalence. We construct an infinite family of genuinely distinct obstacles that prevent an abstract equivalence relation from being a metric-line equivalence.