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

Abstract

The big-line-big-clique conjecture states that for all $k,\ell \ge 2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ 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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