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Erdős-Szekeres-Type Problems in the Real Projective Plane

Authors Martin Balko , Manfred Scheucher , Pavel Valtr

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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Probability and statistics
  • Information systems → Data structures
Keywords
  • real projective plane
  • point set
  • convex position
  • k-gon
  • k-hole
  • Erdős-Szekeres theorem
  • Horton set
  • random point set

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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 Erdős-Szekeres-type problems.
We provide asymptotically tight bounds for a variant of the Erdős-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 Ω(n^{3-3/5k}) 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.

Cite As Get BibTex

Martin Balko, Manfred Scheucher, and Pavel Valtr. Erdős-Szekeres-Type Problems in the Real Projective Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.10

Author Details

Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Funding

  • Balko, Martin: supported by the grant no. 21-32817S of the Czech Science Foundation (GAČR), by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). This article is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).
  • Scheucher, Manfred: supported by the DFG Grant SCHE 2214/1-1.
  • Valtr, Pavel: supported by the grant no. 21-32817S of the Czech Science Foundation (GAČR).

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