A note on visible islands
Sophie Leuchtner, Carlos M. Nicolas, Andrew Suk

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 in the plane, a subset is called an island in if . We say that is a visible island if the points in are pairwise visible and is an island in . The famous Big-line Big-clique Conjecture states that for any and , there is an integer , such that every finite set of at least points in the plane contains collinear points or 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 .
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Commutative Algebra and Its Applications
