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149
Simple proofs of classical theorems in discrete geometry via the GuthKatz polynomial partitioning technique
 DISCRETE COMPUT. GEOM
"... Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well ..."
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Cited by 35 (7 self)
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Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at selfcontained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.
Cutting Circles into Pseudosegments and Improved Bounds for Incidences
 Geom
, 2000
"... We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m ..."
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Cited by 34 (15 self)
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We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree. 1 Introduction Let P be a finite set of points in the plane and C a finite set of circles. Let I = I(P, C) denote the number of incidences between the points and the circles. Let I(m, n) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles, and let I # (m, n, X) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 ...
Extremal Problems for Geometric Hypergraphs
 Discrete Comput. Geom
, 1998
"... A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it ..."
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Cited by 28 (3 self)
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A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the kset problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (isimplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges...
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete a ..."
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Cited by 27 (1 self)
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Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but ..."
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Cited by 25 (6 self)
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We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.
An incidence theorem in higher dimensions
 Discrete Comput. Geom
"... Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points. ..."
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Cited by 23 (0 self)
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Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
The ClarksonShor Technique Revisited and Extended
 Comb., Prob. & Comput
, 2001
"... We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds. ..."
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Cited by 22 (3 self)
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We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds.
The SzemerédiTrotter theorem in the complex plane
, 305
"... This paper generalizes of the SzemerédiTrotter theorem to the complex plane. Szemerédi and Trotter proved that the number of pointline incidences of n points and e lines in the real Euclidean plane is O(n 2/3 e 2/3 + n + e). This bound is tight. Although several short proofs were found to this the ..."
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Cited by 22 (0 self)
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This paper generalizes of the SzemerédiTrotter theorem to the complex plane. Szemerédi and Trotter proved that the number of pointline incidences of n points and e lines in the real Euclidean plane is O(n 2/3 e 2/3 + n + e). This bound is tight. Although several short proofs were found to this theorem [14, 12], and many multidimensional generalizations were given, no tight bound has been known so far for incidences in higher dimensions. We extend the methods of Szemerédi and Trotter and prove that the number of pointline incidences of n points and e complex lines in the complex plane�2 is O(n
Distinct Distances in the Plane
, 2001
"... It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1. ..."
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Cited by 20 (0 self)
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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1.