Erdos-Szekeres-Type Problems in the Real Projective Plane

Martin Balko, Manfred Scheucher, Pavel Valtr

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We consider point sets in the real projective plane RP2 and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdos-Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdos-Szekeres theorem about point sets in convex position in RP2, which was initiated by Harborth and Möller in 1994. The notion of convex position in RP2 agrees with the definition of convex sets introduced by Steinitz in 1913. For k = 3, an (affine) k-hole in a finite set S ? R2 is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from RP2, called projective k-holes, we find arbitrarily large finite sets of points from RP2 with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for k = 7. On the other hand, we show that the number of k-holes can be substantially larger in RP2 than in R2 by constructing, for every k ? (3,..., 6), sets of n points from R2 ? RP2 with ?(n3-3/5k) projective k-holes and only O(n2) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in RP2 and about some algorithmic aspects. The study of extremal problems about point sets in RP2 opens a new area of research, which we support by posing several open problems.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
DOIs
StatePublished - 1 Jun 2022
Externally publishedYes
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume224
ISSN (Print)1868-8969

Conference

Conference38th International Symposium on Computational Geometry, SoCG 2022
Country/TerritoryGermany
CityBerlin
Period7/06/2210/06/22

Keywords

  • Erdos-Szekeres theorem
  • Horton set
  • convex position
  • k-gon
  • k-hole
  • point set
  • random point set
  • real projective plane

ASJC Scopus subject areas

  • Software

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