Lines, betweenness and metric spaces

TL;DR

This paper proves new lower bounds on the number of lines determined by points in finite metric spaces, extending classical Euclidean results to more general metric and graph-induced spaces.

Contribution

It establishes the first polynomial lower bounds on the number of lines in general finite metric spaces and improves bounds for graph-induced metric spaces.

Findings

At least one line contains all points or there are Ω(√n) lines in any metric space.

In pseudometric betweenness, there are Ω(n^{2/5}) lines.

In graph-induced metric spaces, there are Ω(n^{4/7}) lines or one line contains all points.

Abstract

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space. We prove that in any metric space with $n$ points, either there is a line containing all the points or there are at least $\Omega(\sqrt{n})$ lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of $\Omega(n^{2/5})$ lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are $\Omega(n^{4/7})$ lines, improving the previous $\Omega(n^{2/7})$ bound. We also prove that the number of lines in an $n$-point metric space is at least $n / 5w$, where $w$ is the number of different distances in the space, and we give an $\Omega(n^{4/3})$ lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from \{1, 2, 3\}.