2022/06/07 by Sophie Leuchtner, Carlos M. Nicolás, Andrew Suk · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #graph theory and CDMA systems #Rings, Modules, and Algebras
paper · doi:10.1556/012.2022.01524
openalex publication_date 2022/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S . We say that S ⊂ 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 ≥ 3 and l ≥ 4, there is an integer n = n(k, l ), such that every finite set of at least n points in the plane contains l 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.