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arXiv · 1812.06288

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

Vašek Chvátal
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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.

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Vašek Chvátal. 2018-12-15. A De Bruijn-Erd\H{o}s theorem in graphs?. https://doi.org/10.1007/978-3-319-97686-0

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