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17
On the Number of Incidences Between Points and Curves
 Combinatorics, Probability and Computing 7
"... We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane. ..."
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We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane.
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 13 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Simple proofs of classical theorems in discretegeometryviatheGuthKatzpolynomialpartitioningtechnique
 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 8 (4 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. 1
Isosceles Triangles Determined By a Planar Point Set
"... It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5 ..."
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Cited by 8 (2 self)
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It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5e 1 , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi{C. Toth and Tardos.
Some scalable parallel algorithms for geometric problems
 Journal of Parallel and Distributed Computing
, 1999
"... This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary ..."
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Cited by 6 (2 self)
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This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scalable parallel algorithms for a number of geometric problems including the rectangle finding problem, the maximal equally spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scalable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scalability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.
Point Sets with Distinct Distances
, 1995
"... For positive integers d and n let fd(n) denote the maximum cardinality of a subset of the n d grid f1; 2; : : : ; ng d with distinct mutual euclidean distances. Improving earlier results of Erdos and Guy, it will be shown that f2(n) c \Delta n 2=3 and, for d 3, that fd(n) cd \Delta n 2=3 ..."
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Cited by 4 (1 self)
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For positive integers d and n let fd(n) denote the maximum cardinality of a subset of the n d grid f1; 2; : : : ; ng d with distinct mutual euclidean distances. Improving earlier results of Erdos and Guy, it will be shown that f2(n) c \Delta n 2=3 and, for d 3, that fd(n) cd \Delta n 2=3 \Delta (ln n) 1=3 , where c; cd ? 0 are constants. Also improvements of lower bounds of Erdos and Alon on the size of Sidonsets in f1 2 ; 2 2 ; : : : ; n 2 g are given. Furthermore, it will be proven that any set of n points in the plane contains a subset with distinct mutual distances of size c1 \Delta n 1=4 , and for point sets in general position, i.e. no three points on a line, of size c2 \Delta n 1=3 with constants c1 ; c2 ? 0. To do so, it will be shown that for n points in R 2 with distinct distances d1 ; d2 ; : : : ; d t , where d i has multiplicity m i , one has P t i=1 m 2 i c \Delta n 3:25 for a positive constant c. If the n points are in general position,...
On Combinatorics of Projective Mappings
 J. Alg. Combin
, 1998
"... We consider composition sets of onedimensional projective mappings and prove that small composition sets are closely related to Abelian subgroups. 0 Introduction Freiman [5, 6] and Ruzsa [12, 13] studied subsets of R, for which jA+Bj Cn, where jAj = jBj = n. (They described the structure of A a ..."
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We consider composition sets of onedimensional projective mappings and prove that small composition sets are closely related to Abelian subgroups. 0 Introduction Freiman [5, 6] and Ruzsa [12, 13] studied subsets of R, for which jA+Bj Cn, where jAj = jBj = n. (They described the structure of A and B in terms of some natural generalizations of arithmetic progressions.) Using their theorems, BalogSzemer'edi [1] and LaczkovichRuzsa [8] found some "statistical" versions. Their results extend to torsionfree Abelian groups, as well. Generalizations to nonAbelian groups were initiated by the first named author in [3, 4], where the onedimensional affine group was considered. The goal of this paper is to find similar results for the (still onedimensional) projective group. Throughout this paper P will denote the group of nondegenerate projective mappings of R, i.e., the set of nonconstant linear fractions x 7! ax+b cx+d (where ad \Gamma bc 6= 0), with the composition as the group...