Abstract
We consider point sets in the real projective plane ℝ?² and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on ErdősSzekerestype problems.
We provide asymptotically tight bounds for a variant of the ErdősSzekeres theorem about point sets in convex position in ℝ?², which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝ?² agrees with the definition of convex sets introduced by Steinitz in 1913.
For k ≥ 3, an (affine) khole 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 kholes for points sets from ℝ?², called projective kholes, we find arbitrarily large finite sets of points from ℝ?² with no projective 8holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective kholes for k ≤ 7. On the other hand, we show that the number of kholes can be substantially larger in ℝ?² than in ℝ² by constructing, for every k ∈ {3,… ,6}, sets of n points from ℝ² ⊂ ℝ?² with Ω(n^{33/5k}) projective kholes and only O(n²) affine kholes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝ?² and about some algorithmic aspects.
The study of extremal problems about point sets in ℝ?² opens a new area of research, which we support by posing several open problems.
BibTeX  Entry
@InProceedings{balko_et_al:LIPIcs.SoCG.2022.10,
author = {Balko, Martin and Scheucher, Manfred and Valtr, Pavel},
title = {{Erd\H{o}sSzekeresType Problems in the Real Projective Plane}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {10:110:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772273},
ISSN = {18688969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16018},
URN = {urn:nbn:de:0030drops160182},
doi = {10.4230/LIPIcs.SoCG.2022.10},
annote = {Keywords: real projective plane, point set, convex position, kgon, khole, Erd\H{o}sSzekeres theorem, Horton set, random point set}
}
Keywords: 

real projective plane, point set, convex position, kgon, khole, ErdősSzekeres theorem, Horton set, random point set 
Collection: 

38th International Symposium on Computational Geometry (SoCG 2022) 
Issue Date: 

2022 
Date of publication: 

01.06.2022 