The Big-Line-Big-Clique Conjecture is False for Infinite Point Sets

2010/08/18 by David R. Wood, Attila~Pór, Wood, David R.
Mathematics · #52C10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:52C10

paper · pdf · doi:10.48550/arxiv.1008.2988

arxiv created 2010/08/18 · arxiv updated 2010/08/19

Abstract

The big-line-big-clique conjecture states that for all k,ℓ≥2 there is an integer n such that every finite set of at least n points in the plane contains ℓ collinear points or k pairwise visible points. We show that this conjecture is false for infinite point sets, by constructing a countably infinite point set with no 4 collinear points and no 3 pairwise visible points.

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