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

TL;DR

This paper demonstrates that the big-line-big-clique conjecture, which relates to the existence of large collinear or mutually visible point subsets, does not hold for infinite point sets by providing a counterexample.

Contribution

The authors construct a countably infinite point set with no four collinear points and no three mutually visible points, disproving the conjecture for infinite sets.

Findings

Counterexample for infinite point sets

No 4 collinear points in the constructed set

No 3 mutually visible points in the constructed set

Abstract

The big-line-big-clique conjecture states that for all $k,\ell\geq2$ 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.