Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 59: Issue 2
A Note on Visible Islands
Authors:
Sophie Leuchtner University of California at San Diego, La Jolla, CA, 92093 USA

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Carlos M. Nicolás Department of Mathematics, Virginia Tech., Blacksburg, VA, 24061 USA

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Andrew Suk Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093 USA

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Pages:
160–163
Online Publication Date:
07 Jun 2022
Publication Date:
20 Jun 2022
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2022.01524
Keywords:
Visible islands; Horton sets; Big-line Big-clique conjecture
Restricted access

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.

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    A. Abel , B. Ballinger , P. Bose , S. Collette , V. Dujmovic , F. Hurtado , S. D. Kominers , S. Lan-german , A. Pór and D. Wood . Every large point set contains many collinear points or an empty pentagon. Graphs and Combinatorics, 27:47-60, 2011.

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    J. D. Horton . Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482-484, 1983.

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    J. Kára , A. Pór and D. Wood . On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom., 34:497-506, 2005.

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  • [4]

    J. Matousek . Blocking Visibility for Points in General Position. Discrete Comput Geom, 42:219-223, 2009.

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    J. Matousek . Lectures on Discrete Geometry. Springer-Verlag, New York, 2002.

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    A. Pór and D. Wood . On visibility and blockers. J. Comput. Geom., 1:29-40, 2010.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

Editorial Board

  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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