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51
Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
On the number of sums and products
, 1997
"... In what follows A will always denote a finite subset of the nonzero reals, and n the number of its elements. As usual, A + A and A · A stand for the sets of all pairwise sums {a + a ′ : a, a ′ ∈ A} and products {a · a ′ : a, a ′ ∈ A}, respectively. Also, S  denotes the size of a set S. The fo ..."
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Cited by 74 (0 self)
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In what follows A will always denote a finite subset of the nonzero reals, and n the number of its elements. As usual, A + A and A · A stand for the sets of all pairwise sums {a + a ′ : a, a ′ ∈ A} and products {a · a ′ : a, a ′ ∈ A}, respectively. Also, S  denotes the size of a set S. The following problem was posed by Erdős and Szemerédi (see [5]): For a given n, how small can one make A + A  and A · A  simultaneously? In other words, defining a lower estimate should be found for m(A): = max{A + A, A · A}, g(n): = min A=n m(A). R e m a r k. The philosophy behind the question is that either of A + A or A · A  is easy to minimize—just take an arithmetic or geometric (i.e., exponential) progression for A. However, in both of these examples, the other set becomes very large. In their above mentioned paper, Erdős and Szemerédi managed to prove the existence of a small but positive constant c1 such that g(n) ≥ n 1+c1 for all n. (See also p. 107 of Erdős ’ paper [3].) Later on, Nathanson and K. Ford found the lower bounds n 32/31 and n 16/15, respectively [7]. The goal of this paper is to improve the exponent to 5/4. Theorem 1. There is a positive absolute constant c such that, for every nelement set A, c · n 5/4 ≤ max{A + A, A · A}.
Distinct distances in high dimensional homogeneous sets, Towards a theory of geometric graphs
, 2004
"... We show that the number of distinct distances in a welldistributed set of n points in Rd is Ω(n2/d−1/d2) which is not far from the best known upper bound O(n2/d). For the special case d = 3, we have a further improvement Ω(n.5794). 1 ..."
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We show that the number of distinct distances in a welldistributed set of n points in Rd is Ω(n2/d−1/d2) which is not far from the best known upper bound O(n2/d). For the special case d = 3, we have a further improvement Ω(n.5794). 1
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 21 (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.
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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Cited by 14 (0 self)
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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...
Counting Facets and Incidences
 DISCRETE COMPUT GEOM 7:359369 (1992)
, 1992
"... We show that m distinct cells in an arrangement of n planes in R a are bounded by O(m2/an + n 2) faces, which in turn yields a tight bound on the maximum number of facets bounding m cells in an arrangement of n hyperplanes in R a, for every d> 3. In addition, the method is extended to obtain tigh ..."
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Cited by 13 (3 self)
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We show that m distinct cells in an arrangement of n planes in R a are bounded by O(m2/an + n 2) faces, which in turn yields a tight bound on the maximum number of facets bounding m cells in an arrangement of n hyperplanes in R a, for every d> 3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in R 3. We also present a simpler proof of the O(m2/3n ~/3 + n d 1) bound on the number of incidences between n hyperplanes in R a and m vertices of their arrangement.
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 l ..."
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Cited by 13 (3 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.