Crossing Numbers and Hard Erdős Problems in Discrete Geometry

Székely, László Á.

doi:10.1017/s0963548397002976

Cited by 262 publications

(246 citation statements)

References 21 publications

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“…We can now continue by applying the crossing lemma argument, exactly as done by Székely and in other works (e.g., see [8,13]), and conclude that…”

Section: Proof Of Propositionmentioning

confidence: 62%

“…This is an instance of a fairly standard point-curve incidence problem, which can be tackled using the well established machinery, such as the incidence bound of Pach and Sharir [8], or, more fundamentally, the crossing-lemma technique of Székely [13] (on which the analysis in [8] is based). However, to apply this machinery, it is essential that the curves of Γ have a constant bound on their multiplicity.…”

Section: Unit Circles Spanned By Points On Three Unit Circlesmentioning

confidence: 99%

“…We apply Székely's technique [13], which is based on the crossing lemma (see also Pach and Agarwal [7]). As noted, this is also the approach used in [8], but the possible overlap of curves requires some extra (and more explicit) care in the application of the technique.…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 2 more Smart Citations

On Triple Intersections of Three Families of Unit Circles

Raz

Sharir

Solymosi

2015

Discrete Comput Geom

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Let p 1 , p 2 , p 3 be three distinct points in the plane, and, for i = 1, 2, 3, let C i be a family of n unit circles that pass through p i . We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n 11/6 ), improving an earlier bound for this problem due to Elekes, Simonovits, and Szabó [4]. The problem is a special instance of a more general problem studied by Elekes and Szabó [5] (and by Elekes and Rónyai [3]).

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“…We can now continue by applying the crossing lemma argument, exactly as done by Székely and in other works (e.g., see [8,13]), and conclude that…”

Section: Proof Of Propositionmentioning

confidence: 62%

Section: Unit Circles Spanned By Points On Three Unit Circlesmentioning

confidence: 99%

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On Triple Intersections of Three Families of Unit Circles

Raz

Sharir

Solymosi

2015

Discrete Comput Geom

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“…This result implied an upper bound of O(n log 3 n log log n) on the running time of Efrat and Sharir's [8] segment center algorithm. Pach and Tardos [28] used the forbidden submatrix framework to obtain a new proof that there are at most O(n 4/3 ) unit distances among n points in the plane, matching the best known upper bound [31,23,32,1]. Very recently the author [30] has shown that numerous data structures based on path compression and binary search trees can be analyzed in a simple, uniform way using the forbidden submatrix method.…”

Section: Introductionmentioning

confidence: 99%

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

Pettie

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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A 0-1 matrix A is said to avoid a forbidden 0-1 matrix (or pattern) P if no submatrix of A matches P , where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport-Schinzel sequences and their generalizations, Stanley and Wilf's permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0-1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Our primary contribution is a refutation of a conjecture of Füredi and Hajnal: that if P corresponds to an acyclic graph then the maximum number of 1s in an n × n matrix avoiding P is O(n log n). In addition, we give a simpler proof that there are infinitely many minimal nonlinear patterns and give tight bounds on the extremal functions for several small forbidden patterns.

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“…This result was used to deduce the best known lower bound for the number of distinct distances determined by n points in the plane [Sz95], [ST01], [KT04] and upper bound for the number of different ways how a line can split a set of 2n points into two equal parts [D98], and it has some other interesting corollaries [PS98], [PT02], [STT02], [MSSW06], [BCSV07].…”

Section: Introductionmentioning

confidence: 99%

Crossing numbers of imbalanced graphs

Pach

Solymosi

Tardos

2009

Journal of Graph Theory

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The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e 3 /n 2 . Apart from the value of the constant, this bound cannot be improved. We establish some stronger lower bounds, under the assumption that the distribution of the degrees of the vertices is irregular. In particular, we show that if the degrees of the vertices are d1 ≥ d2 ≥ . . . ≥ dn, then the crossing number satisfies cr(G) ≥ c 1 n n i=1 id 3 i − c2n 2 , and that this bound is tight apart from the values of the constants c1, c2 > 0. Some applications are also presented.

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Crossing Numbers and Hard Erdős Problems in Discrete Geometry

Cited by 262 publications

References 21 publications

On Triple Intersections of Three Families of Unit Circles

On Triple Intersections of Three Families of Unit Circles

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

Crossing numbers of imbalanced graphs

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