A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n 4/3 ) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let La and L b be two sets of t lines in the plane and let P = { a ∩ b : a ∈ La, b ∈ L b } be the set of intersection points between La and L b . We say that (P, La ∪ L b ) forms a natural t × t grid if |P | = t 2 , and conv(P ) does not contain the intersection point of some two lines in La and does not contain the intersection point of some two lines in L b . For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t × t grid determines O(n 4 3 −ε ) incidences, where ε = ε(t) > 0. We also provide a construction of n points and n lines in the plane that does not contain a natural 2 × 2 grid and determines at least Ω(n 1+ 1 14 ) incidences.

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“…We also note that a similar result was established by Dujmović and Langerman (see Theorem 6 in [6]).…”

Section: Discussionsupporting

confidence: 85%

On Grids in Point-Line Arrangements in the Plane

Mirzaei

Suk

2020

Discrete Comput Geom

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“…The main difference is that we use the result of Fox et al [8] instead of the Ham-Sandwich Theorem. We also note that a similar result was established by Dujmović and Langerman (see Theorem 6 in [6]).…”

Section: Discussionsupporting

confidence: 85%

On grids in point-line arrangements in the plane

Mirzaei¹,

Suk²

2018

Preprint

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The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n 4/3 ) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let L 1 and L 2 be two sets of t lines in the plane and let P = { 1 ∩ 2 : 1 ∈ L 1 , 2 ∈ L 2 } be the set of intersection points between L 1 and L 2 . We say that (P, L 1 ∪ L 2 ) forms a natural t × t grid if |P | = t 2 , and conv(P ) does not contain the intersection point of some two lines in L i , for i = 1, 2. For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t × t grid determines O(n 4 3 −ε ) incidences, where ε = ε(t). We also provide a construction of n points and n lines in the plane that does not contain a natural 2 × 2 grid and determines at least Ω(n 1+ 1 14 ) incidences.

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“…Dujmović and Langerman [11] used the existence of OSH d (n) to prove a ham-sandwich cut theorem for hyperplanes. Matoušek and Welzl [21] observed that OSH 2 (n) = (n − 1) 2 + 1 and Eliáš and Matoušek [12] noticed that, for d ≥ 3, OSH d (n) ≤ twr d (cn) follows from Ramsey's theorem, where c depends on d. In Section 4, we prove the following result which again improves the upper bound by one exponential.…”

Section: Introductionmentioning

confidence: 99%

Ramsey-type results for semi-algebraic relations

Conlon

Fox

Pach

et al. 2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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For natural numbers d and t there exists a positive C such that if F is a family of n C semialgebraic 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, Radoičić, and Sharir and improves on a bound of 4 n 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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A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Cited by 13 publications

References 20 publications

On Grids in Point-Line Arrangements in the Plane

On Grids in Point-Line Arrangements in the Plane

On grids in point-line arrangements in the plane

Ramsey-type results for semi-algebraic relations

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