Lines in quasi-metric spaces with four points

TL;DR

This paper investigates lines in quasi-metric spaces with four points, proving a specific betweenness structure exists and confirming Chen and Chvátal's conjecture holds for metric spaces on four points.

Contribution

It constructs a unique betweenness structure on four points in a quasi-metric space and shows the conjecture's validity for metric spaces with four points.

Findings

Existence of a specific betweenness structure on four points

Quasi-metric space with only three lines, none with four points

Chen and Chvátal's conjecture holds for four-point metric spaces

Abstract

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'atal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. In this work we prove that there is a quasi-metric space on four points a, b, c and d whose betweenness is B={(c,a,b),(a,b,c),(d,b,a),(b,a,d)}. Then, this space has only three lines none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we get that Chen and Chv\'atal's conjecture is valid for any metric space on four points.