Towards a de Bruijn-Erd\H os theorem in the $L_1$-metric

TL;DR

This paper extends the de Bruijn-Erdős theorem to finite metric spaces under the $L_1$ metric, proving that sets of points either form a single line or induce many lines, with bounds depending on coordinate sharing.

Contribution

It proves an $L_1$ metric analogue of the de Bruijn-Erdős theorem, establishing conditions for the number of lines determined by points in the plane.

Findings

Either one line contains all points or there are at least n lines when no two points share coordinates.

If points share coordinates, then there are at least n/37 lines.

The result confirms the conjecture for the $L_1$ metric in the plane.

Abstract

A well-known theorem of de Bruijn and Erd\H{o}s states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chv\'{a}tal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of {\em betweenness}. In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$- or $y$-coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines. If points of $X$ are allowed to share their coordinates, then either there is a line that contains all $n$ points, or $X$ induces at least $n/37$ distinct lines.