A note on visible islands

TL;DR

This paper disproves a variant of the Big-line Big-clique Conjecture by constructing large point sets with no small visible islands or collinear points, using Horton sets with triples of collinear points.

Contribution

It demonstrates that the Big-line Big-clique Conjecture does not hold for visible islands by providing a counterexample construction.

Findings

Counterexample with no 4 collinear points

No visible island of size 13 in large sets

Disproves the conjecture for visible islands

Abstract

Given a finite point set $P$ in the plane, a subset $S \subseteq P$ is called an island in $P$ if $conv(S) \cap P = S$. We say that $S\subset P$ is a visible island if the points in $S$ are pairwise visible and $S$ is an island in $P$. The famous Big-line Big-clique Conjecture states that for any $k \geq 3$ and $\ell \geq 4$, there is an integer $n = n(k,\ell)$, such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size $13$.