Erdős-Szekeres-Type Problems in the Real Projective Plane

Authors Martin Balko , Manfred Scheucher , Pavel Valtr

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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

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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)


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.

Subject Classification

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


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  1. O. Aichholzer, F. Aurenhammer, and H. Krasser. Data base of order types for small point sets. URL:
  2. O. Aichholzer, F. Aurenhammer, and H. Krasser. Enumerating order types for small point sets with applications. Order, 19(3):265-281, 2002. URL:
  3. O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber. A superlinear lower bound on the number of 5-holes. Journal of Combinatorial Theory, Series A, 173:Paper No. 105236, 2020. URL:
  4. B. Aronov, P. Erdős, W. Goddard, D. J. Kleitman, M. Klugerman, J. Pach, and L. J. Schulman. Crossing families. Combinatorica, 14(2):127-134, 1994. Google Scholar
  5. M. Balko, M. Scheucher, and P. Valtr. Holes and islands in random point sets. Random Structures & Algorithms, 2021. URL:
  6. M. Balko, M. Scheucher, and P. Valtr. Tight bounds on the expected number of holes in random point sets, 2021. URL:
  7. M. Balko, M. Scheucher, and P. Valtr. Erdős-Szekeres-type problems in the real projective plane, 2022. URL:
  8. I. Bárány and Z. Füredi. Empty simplices in Euclidean space. Canadian Mathematical Bulletin, 30(4):436-445, 1987. URL:
  9. I. Bárány and P. Valtr. Planar point sets with a small number of empty convex polygons. Studia Scientiarum Mathematicarum Hungarica, 41(2):243-266, 2004. URL:
  10. J. Bracho and G. Calvillo. Homotopy classification of projective convex sets. Geometriae Dedicata, 37:303-306, 1991. URL:
  11. B. Bukh, T. Chao, and R. Holzman. On convex holes in d-dimensional point sets, 2020. URL:
  12. D. Conlon and J. Lim. Fixing a hole., 2021. URL:
  13. J. de Groot and H. de Vries. Convex sets in projective space. Compositio Mathematica, 13:113-118, 1958. URL:
  14. D. Dekker. Convex regions in projective n-space. The American Mathematical Monthly, 62(6):430-431, 1955. Google Scholar
  15. P. Erdős. Some more problems on elementary geometry. Australian Mathematical Society Gazette, 5:52-54, 1978. URL:
  16. P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. URL:
  17. P. Erdős and G. Szekeres. On some extremum problems in elementary geometry. Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica, 3-4:53-63, 1960. Google Scholar
  18. R. Fabila-Monroy and C. Huemer. Covering Islands in Plane Point Sets. In Computational Geometry: XIV Spanish Meeting on Computational Geometry, EGC 2011, volume 7579 of Lecture Notes in Computer Science, pages 220-225. Springer, 2012. URL:
  19. L. Finschi and K. Fukuda. Generation of oriented matroids-a graph theoretical approach. Discrete & Computational Geometry, 27(1):117-136, 2002. URL:
  20. Lukas Finschi. Webpage: Homepage of oriented matroids. URL:
  21. T. Gerken. Empty Convex Hexagons in Planar Point Sets. Discrete & Computational Geometry, 39(1):239-272, 2008. URL:
  22. P. Giannopoulos, C. Knauer, and D. Werner. On the computational complexity of Erdős-Szekeres and related problems in ℝ³. In Algorithms - ESA 2013, pages 541-552. Springer, 2013. URL:
  23. B. P. Haalmeyer. Bijdragen tot de theorie der elementairoppervlakken. PhD thesis, Amsterdam, 1917. Google Scholar
  24. H. Harborth and M. Möller. The Esther-Klein-problem in the projective plane. Journal of Combinatorial Mathematics and Combinatorial Computing, 15, 1993. Google Scholar
  25. A. F. Holmsen, H. N. Mojarrad, J. Pach, and G. Tardos. Two extensions of the Erdős-Szekeres problem. Journal of the European Mathematical Society, 22:3981-3995, 2020. URL:
  26. J. D. Horton. Sets with no empty convex 7-gons. Canadian Mathematical Bulletin, 26:482-484, 1983. URL:
  27. F. Hurtado, M. Noy, and J. Urrutia. Flipping edges in triangulations. Discrete & Computational Geometry, 22(3):333-346, 1999. URL:
  28. H. Kneser. Eine Erweiterung des Begriffes "konvexer Körper". Mathematische Annalen, 82(3):287-296, 1921. In German. URL:
  29. F. Marić. Fast formal proof of the Erdős-Szekeres conjecture for convex polygons with at most 6 points. Journal of Automated Reasoning, 62:301-329, 2019. URL:
  30. J. S. B. Mitchell, G. Rote, G. Sundaram, and G. Woeginger. Counting convex polygons in planar point sets. Information Processing Letters, 56(1):45-49, 1995. URL:
  31. C. M. Nicolas. The empty hexagon theorem. Discrete & Computational Geometry, 38(2):389-397, 2007. URL:
  32. J. Pach, N. Rubin, and G. Tardos. Planar point sets determine many pairwise crossing segments. Advances in Mathematics, 386:Paper No. 107779, 2021. URL:
  33. M. Scheucher. Two disjoint 5-holes in point sets. Computational Geometry, 91:Paper No. 101670, 2020. URL:
  34. M. Scheucher. A SAT attack on higher dimensional Erdős-Szekeres numbers. In Extended Abstracts EuroComb 2021, pages 103-110. Springer, 2021. URL:
  35. E. Steinitz. Bedingt konvergente Reihen und konvexe Systeme. Journal für die reine und angewandte Mathematik, 143:128-176, 1913. In German. URL:
  36. A. Suk. On the Erdős-Szekeres convex polygon problem. Journal of the AMS, 30:1047-1053, 2017. URL:
  37. G. Szekeres and L. Peters. Computer solution to the 17-point Erdős-Szekeres problem. Australia and New Zealand Industrial and Applied Mathematics, 48(2):151-164, 2006. URL:
  38. P. Valtr. Convex independent sets and 7-holes in restricted planar point sets. Discrete & Computational Geometry, 7(2):135-152, 1992. URL:
  39. P. Valtr. Sets in ℝ^d with no large empty convex subsets. Discrete Mathematics, 108(1):115-124, 1992. URL:
  40. P. Valtr. On the minimum number of empty polygons in planar point sets. Studia Scientiarum Mathematicarum Hungarica, pages 155-163, 1995. URL: