De Bruijn-Erd\H{o}s type theorems for graphs and posets

TL;DR

This paper extends the classical De Bruijn-Erdős theorem from geometry to graphs and posets, establishing minimum line counts and extremal configurations, with improved bounds for certain posets.

Contribution

It introduces graph and poset analogues of the De Bruijn-Erdős theorem, including bounds based on poset height and characterization of extremal cases.

Findings

Graphs satisfy a De Bruijn-Erdős type line bound.

Posets, especially comparability graphs, have similar bounds with improvements for height.

Extremal configurations are explicitly characterized.

Abstract

A classical theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to comparability graphs, we obtain a version of the De Bruijn-Erd\H{o}s theorem for partially ordered sets (posets). Moreover, in this case, we have an improved bound on the number of lines depending on the height of the poset. The extremal configurations are also determined.