De Bruijn-Erd\H{o}s type theorems for graphs and posets
Combinatorics
2015-01-29 v2
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.
Cite
@article{arxiv.1501.06681,
title = {De Bruijn-Erd\H{o}s type theorems for graphs and posets},
author = {Pierre Aboulker and Guillaume Lagarde and David Malec and Abhishek Methuku and Casey Tompkins},
journal= {arXiv preprint arXiv:1501.06681},
year = {2015}
}