research-article

Higher-order Erdös: Szekeres theorems

Authors:

Marek Eliáý,

Jiõí MatouýekAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 81 - 90

https://doi.org/10.1145/2261250.2261264

Published: 17 June 2012 Publication History

Get Access

Abstract

Let P=(p₁,p₂,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x₁<x₂...x_N. A famous 1935 Erdos-Szekeres theorem asserts that every such P contains a monotone subsequence S of √N points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Ω(log N) points.

Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k+1)-tuple K ⊆ P to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k+1)-tuple. Then we say that S ⊆ P is kth-order monotone if its (k+1)-tuples are all positive or all negative.

We investigate quantitative bound for the corresponding Ramsey-type result (i.e., how large kth-order monotone subsequence can be guaranteed in every N-point P). We obtain an Ω(log^(k-1)N) lower bound ((k-1)-times iterated logarithm). This is based on a quantitative Ramsey-type theorem for what we call transitive colorings of the complete (k+1)-uniform hypergraph; it also provides a unified view of the two classical Erdos-Szekeres results mentioned above.

For k=3, we construct a geometric example providing an O(log log N) upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for order-type homogeneous subsets in R³, as well as for a Ramsey-type theorem for hyperplanes in R⁴ recently used by Dujmovic and Langerman.

References

[1]

D. Conlon, J. Fox, and B. Sudakov. An improved bound for the stepping-up lemma. Discrete Applied Mathematics, 2011. In press.

Digital Library

Google Scholar

[2]

V. Dujmovic and S. Langerman. A center transversal theorem for hyperplanes and applications to graph drawing. In Proc. 27th ACM Symposium on Computational Geometry, pages 117--124, 2011. Full version arXiv:1012.0548.

Digital Library

Google Scholar

[3]

P. Erdos and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc., 3:417--439, 1952.

Crossref

Google Scholar

[4]

P. Erdos and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2:463--470, 1935.

Google Scholar

[5]

J. Fox, J. Pach, B. Sudakov, and A. Suk. Erdos-Szekeres-type theorem for monotone paths and convex bodies. Arxiv preprint 1105.2097v1, 2011. Proc. London Math. Soc., in press.

Google Scholar

[6]

J. E. Goodman and R. Pollack. Allowable sequences and order types in discrete and computational geometry. In J. Pach, editor, New Trends in Discrete and Computational Geometry, volume 10 of Algorithms and Combinatorics, pages 103--134. Springer, Berlin etc., 1993.

Google Scholar

[7]

R. L. Graham, B. L. Rothschild, and J. Spencer. Ramsey Theory. J. Wiley & Sons, New York, 1990.

Crossref

Google Scholar

[8]

J. Matousek. Lectures on Discrete Geometry. Springer, New York, 2002.

Digital Library

Google Scholar

[9]

W. Morris and V. Soltan. The Erdos--Szekeres problem on points in convex position--survey. Bull. Amer. Math. Soc., New Ser., 37(4):437--458, 2000.

Crossref

Google Scholar

[10]

J. Matousek and Emo Welzl. Good splitters for counting points in triangles. J. Algorithms, 13:307--319, 1992.

Digital Library

Google Scholar

[11]

George M. Phillips. Interpolation and approximation by polynomials. Springer, Berlin etc., 2003.

Google Scholar

[12]

G. Rote. Private communication, February 2012.

Google Scholar

[13]

M. J. Steele. Variations on the monotone subsequence theme of Erdos and Szekeres. In D. Aldous et al., editors, Discrete Probability and Algorithms, IMA Volumes in Mathematics and its Applications 72, pages 111--131. Springer, Berlin etc., 1995.

Google Scholar

[14]

P. Valtr. Open caps and cups in planar point sets. Discr. Comput. Geom., 37:365--567, 2004.

Digital Library

Google Scholar

Cited By

View all

Conlon DFox JPach JSudakov BSuk Ada Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Ramsey-type results for semi-algebraic relationsProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462399(309-318)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462399

Index Terms

Higher-order Erdös: Szekeres theorems
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics

Recommendations

Erdös-Szekeres theorem with forbidden order types

According to the classical Erdös-Szekeres theorem, every sufficiently large set of points in general position in the plane contains a large subset in convex position. Parallel to the Erdös-Hajnal problem in graph-Ramsey theory, we investigate how large ...
Convex hulls of spheres and convex hulls of disjoint convex polytopes

Given a set @S of spheres in E^d, with d>=3 and d odd, having a constant number of m distinct radii @r"1,@r"2,...,@r"m, we show that the worst-case combinatorial complexity of the convex hull of @S is @Q(@__ __"1"=<"i"<>"j"=<"mn"in"j^@__ __^d^2^@__ __), ...
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

Given a set Σ of spheres in E^d, with d≥3 and d odd, having a fixed number of m distinct radii ρ₁,ρ₂,...,ρ_m, we show that the worst-case combinatorial complexity of the convex hull CH_d(Σ) of Σ is Θ(Σ_{{1≤i≠j≤m}}n_in_j^{⌊ d/2 ⌋}), where n_i is the number of ...

Comments

Information & Contributors

Information

Published In

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2012

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

SoCG '12

Sponsor:

SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

1
Total Citations
View Citations
100
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 08 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Conlon DFox JPach JSudakov BSuk Ada Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Ramsey-type results for semi-algebraic relationsProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462399(309-318)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462399

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Abstract

References

Cited By

Index Terms

Recommendations

Erdös-Szekeres theorem with forbidden order types

Convex hulls of spheres and convex hulls of disjoint convex polytopes

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Comments

Information

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Conference

Acceptance Rates

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

Login options

Full Access

View options

PDF

eReader

Share

Share this Publication link

Share on social media

Affiliations