A de Bruijn - Erd\H{o}s theorem and metric spaces

TL;DR

This paper explores a generalization of the de Bruijn-Erdős theorem from Euclidean geometry to metric spaces, providing partial results towards understanding the minimum number of lines determined by points in such spaces.

Contribution

It offers initial partial results extending the de Bruijn-Erdős theorem to metric spaces, a novel direction in combinatorial geometry.

Findings

Partial validation of the generalized theorem in specific metric spaces

Insights into the structure of lines in metric spaces

Foundation for further research in metric space geometry

Abstract

De Bruijn and Erd\H{o}s proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chv\'atal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.