TY - GEN

T1 - Erdos-Szekeres-Type Problems in the Real Projective Plane

AU - Balko, Martin

AU - Scheucher, Manfred

AU - Valtr, Pavel

N1 - Publisher Copyright:
© Martin Balko, Manfred Scheucher, and Pavel Valtr; licensed under Creative Commons License CC-BY 4.0

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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.

AB - 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.

KW - Erdos-Szekeres theorem

KW - Horton set

KW - convex position

KW - k-gon

KW - k-hole

KW - point set

KW - random point set

KW - real projective plane

UR - http://www.scopus.com/inward/record.url?scp=85134333189&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2022.10

DO - 10.4230/LIPIcs.SoCG.2022.10

M3 - Conference contribution

AN - SCOPUS:85134333189

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 38th International Symposium on Computational Geometry, SoCG 2022

A2 - Goaoc, Xavier

A2 - Kerber, Michael

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 38th International Symposium on Computational Geometry, SoCG 2022

Y2 - 7 June 2022 through 10 June 2022

ER -