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24
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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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
SylvesterGallai theorem and metric betweenness
, 2002
"... Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines fro ..."
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Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the SylvesterGallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points of the space. Then we present slight evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness. 1 The SylvesterGallai theorem Sylvester (1893) proposed the following problem: Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all line in the same right line.
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...
Cardinalities Of KDistance Sets In Minkowski Spaces
, 1997
"... A subset of a metric space is a kdistance set if there are exactly k nonzero distances occuring between points. We conjecture that a kdistance set in a ddimensional Banach space (or Minkowski space), contains at most (k + 1) points, with equality i the unit ball is a parallelotope. We solv ..."
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A subset of a metric space is a kdistance set if there are exactly k nonzero distances occuring between points. We conjecture that a kdistance set in a ddimensional Banach space (or Minkowski space), contains at most (k + 1) points, with equality i the unit ball is a parallelotope. We solve this conjecture in the armative for all 2dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we nd various weaker upper bounds for kdistance sets.
The k Most Frequent Distances in the Plane
"... A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is fn (k) = O(n ) improving earlier bound of Akatsu, Tamaki, and ..."
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A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is fn (k) = O(n ) improving earlier bound of Akatsu, Tamaki, and Tokuyama.
Improvement Of Inequalities For The (r,q)Structures And Some Geometrical Connections
, 1995
"... . The main results are the inequalities (1) and (6) for the minimal number of (r; q)structure classes,which improve the ones from [3], and also some geometrical connections, especially the inequality (13). 1. Inequalities 1.1. Definition. Let m; n; r; q be natural numbers such that n 3 and r n. ..."
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. The main results are the inequalities (1) and (6) for the minimal number of (r; q)structure classes,which improve the ones from [3], and also some geometrical connections, especially the inequality (13). 1. Inequalities 1.1. Definition. Let m; n; r; q be natural numbers such that n 3 and r n. Let M be a set which contains at least n+q \Gamma1 elements. Let A = fa 1 ; :::; ang ae M . Let P (M ) be the set of all the subsets of M . Let the set B = fB 1 ; :::; Bmg ae P (M ) fulfil the following three conditions : (i) Each element B k 2 B for k = 1; :::; m contains at least r distinct elements a i 1 ; a i 2 ; :::; a i r 2 A ; (ii) If a i 1 ; a i 2 ; :::; a i r are r distinct elements of the set A, then there exist exactly q distinct elements B j1 ; B j2 ; :::; B jq 2 B such that for p = 1; 2; :::; q we have: a i s 2 B jp for each s 2 f1; 2; :::; rg; (iii) For every r +1 distinct elements a i 1 ; a i 2 ; :::; a i r+1 2 A there are at most one element B k 2 B such that a i s 2 B k...
Radial Points in the Plane
, 2001
"... A radial point for a finite set P in the plane is a point q 62 P with the property that each line connecting q to a point of P passes through at least one other element of P . We prove a conjecture of Pinchasi, by showing that the number of radial points for a noncollinear nelement set P is O(n). ..."
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A radial point for a finite set P in the plane is a point q 62 P with the property that each line connecting q to a point of P passes through at least one other element of P . We prove a conjecture of Pinchasi, by showing that the number of radial points for a noncollinear nelement set P is O(n). We also present several extensions of this result, generalizing theorems of Beck, Szemer'edi and Trotter, and Elekes on the structure of incidences between points and lines. 1 Introduction Let P be a set of n points in the plane, not all lying on the same line. A point q = 2 P is called a radial point (for P ) if for every line ` passing through q we have j` " P j 6= 1. In other words, every line connecting q to some point p 2 P passes through at least one other element of P . For instance, let P be the vertex set of a regular 2kgon in the plane. Then, the intersection of the line at infinity with each line supporting an edge of P is a radial point for P . The center of the regular 2kgon ...
Roetwk. Ma*. Kolloq. B. 6 14 (1969)
"... Problemi and rwulte on extreme1 problas in number theory. aeoaetry, aml mebinatorice During my lcmg life I wrote rany papere abaut solved and uneolved problem. I will start with number theory. 1. Perhaps w first eeriwe qonjecture which goem back to 1931 or 32 etatee a8 follae: Let 1 < al < a2 <... < ..."
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Problemi and rwulte on extreme1 problas in number theory. aeoaetry, aml mebinatorice During my lcmg life I wrote rany papere abaut solved and uneolved problem. I will start with number theory. 1. Perhaps w first eeriwe qonjecture which goem back to 1931 or 32 etatee a8 follae: Let 1 < al < a2 <... < 4, be a eenumce of integers. Aesurs that all the euw ere distinct. Is it then true that there is an absolute constant c? for which I idlatsly prwd by a eirple counting arqumemt that and in 1954 using the eecond mt method Leo Moeer and 1 proved that which ie the current record. Conww and Guy proved that for large n F+, < 2n2 ie possible and it has been conjeotured
On distinct distances in homogeneous sets in the Euclidean space
, 2008
"... A homogeneous set of n points in the ddimensional Euclidean space determines at least Ω(n 2d/(d2 +1) /log c(d) n) distinct distances for a constant c(d)> 0. In threespace, we slightly improve our general bound and show that a homogeneous set of n points determines at least Ω(n.6071) distinct dista ..."
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A homogeneous set of n points in the ddimensional Euclidean space determines at least Ω(n 2d/(d2 +1) /log c(d) n) distinct distances for a constant c(d)> 0. In threespace, we slightly improve our general bound and show that a homogeneous set of n points determines at least Ω(n.6071) distinct distances. 1