research-article

Ramsey-type results for semi-algebraic relations

Authors:

Andrew SukAuthors Info & Claims

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

Pages 309 - 318

https://doi.org/10.1145/2462356.2462399

Published: 17 June 2013 Publication History

Abstract

For natural numbers d and t there exists a positive C such that if F is a family of n^C semi-algebraic sets in R^d of description complexity at most t, then there is a subset F' of F of size $n$ such that either every pair of elements in F' intersect or the elements of F' are pairwise disjoint. This result, which also holds if the intersection relation is replaced by any semi-algebraic relation of bounded description complexity, was proved by Alon, Pach, Pinchasi, Radoicic, and Sharir and improves on a bound of 4ⁿ for the family F which follows from a straightforward application of Ramsey's theorem. We extend this semi-algebraic version of Ramsey's theorem to k-ary relations and give matching upper and lower bounds for the corresponding Ramsey function, showing that it grows as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential. We apply this result to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.

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Cited By

Eliáš MMatoušek JRoldán-Pensado ESafernová ZCheng SDevillers O(2014)Lower bounds on geometric Ramsey functionsProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582146(558-564)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582146
Bárány IMatoušek JPór ACheng SDevillers O(2014)Curves in Rd intersecting every hyperplane at most d + 1 timesProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582132(565-571)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582132
Eliáš MMatoušek J(2013)Higher-order Erdős–Szekeres theoremsAdvances in Mathematics10.1016/j.aim.2013.04.020244(1-15)Online publication date: Sep-2013
https://doi.org/10.1016/j.aim.2013.04.020

Index Terms

Ramsey-type results for semi-algebraic relations
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

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cover image ACM Conferences

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

June 2013

472 pages

ISBN:9781450320313

DOI:10.1145/2462356

General Chairs:
Guilherme D. da Fonseca
UNIRIO
,
Thomas Lewiner
PUC-Rio
,
Luis Peñaranda
IMPA
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Program Chairs:
Timothy Chan
University of Waterloo
,
Rolf Klein
University of Bonn

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Published: 17 June 2013

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June 17 - 20, 2013

Rio de Janeiro, Brazil

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SoCG '13 Paper Acceptance Rate 48 of 137 submissions, 35%;

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Cited By

Eliáš MMatoušek JRoldán-Pensado ESafernová ZCheng SDevillers O(2014)Lower bounds on geometric Ramsey functionsProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582146(558-564)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582146
Bárány IMatoušek JPór ACheng SDevillers O(2014)Curves in Rd intersecting every hyperplane at most d + 1 timesProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582132(565-571)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582132
Eliáš MMatoušek J(2013)Higher-order Erdős–Szekeres theoremsAdvances in Mathematics10.1016/j.aim.2013.04.020244(1-15)Online publication date: Sep-2013
https://doi.org/10.1016/j.aim.2013.04.020

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