# A De Bruijn-Erd\H{o}s theorem in graphs?

## TL;DR

This paper explores a De Bruijn-Erdős type theorem in general metric spaces, focusing on graphs, and discusses related open problems and conjectures in the field.

## Contribution

It investigates the validity of a De Bruijn-Erdős theorem in metric spaces derived from graphs and highlights numerous open problems and conjectures.

## Key findings

- The theorem remains unresolved in graph metric spaces.
- Identifies 29 open problems related to the theorem.
- Proposes 3 new conjectures in the area.

## Abstract

A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chv\'atal asked whether the same statement holds true in general metric spaces, where the line determined by points $x$ and $y$ is defined as the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two. The conjecture that it does hold true remains unresolved even in the special case where the metric space arises from a connected undirected graph with unit lengths assigned to edges. We trace its curriculum vitae and point out twenty-nine related open problems plus three additional conjectures.