A De Bruijn-Erdos theorem for 1-2 metric spaces

TL;DR

This paper extends a classical combinatorial geometry theorem to certain metric spaces, proving that a lower bound on the number of lines determined by points holds when distances are only 1 or 2.

Contribution

It establishes a De Bruijn-Erdos type theorem for metric spaces with distances of 1 or 2, broadening the theorem's applicability beyond Euclidean spaces.

Findings

Proves the generalized theorem for metric spaces with distances 1 or 2.

Shows that such metric spaces determine at least n lines for n points.

Extends classical geometric combinatorics to discrete metric spaces.

Abstract

A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.