Betweenness structures of small linear co-size

One way to study the combinatorics of finite metric spaces is betweenness relation associated with space. In hypergraph metrization problem, one has find and characterize betweennesses collinear triples (or alternatively, non-degenerate triangles) that coincide edges a given 3-uniform hypergraph. Metrizability different kinds hypergraphs was investigated in last decades. Chen showed Steiner triple systems are not metric, while Richmond characterized linear betweennesses, i.e. realize complete The latter result also generalized pseudometric (almost-metric) by Beaudou et al. this paper, we further extend theory characterizing largest non-linear almost-metric satisfy certain hereditary properties, as well ones contain small number triangles.