Results 1  10
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24
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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Cited by 113 (1 self)
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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.
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 16 (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.
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 tight b ..."
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Cited by 11 (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 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.
On the number of directions determined by a threedimensional points set
 J. Combin. Theory Ser. A
"... Let P be a set of n points in R 3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n − 3 different directions if n is even and at least 2n − 2 if n is odd. These bounds are sharp. The pr ..."
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Cited by 7 (2 self)
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Let P be a set of n points in R 3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n − 3 different directions if n is even and at least 2n − 2 if n is odd. These bounds are sharp. The proof is based on a farreaching generalization of Ungar’s theorem concerning the analogous problem in the plane. 1
Incidences of nottoodegenerate hyperplanes
 IN PROC. 21ST
, 2005
"... We present a multidimensional generalization of the SzemerédiTrotter Theorem, and give asharp bound on the number of incidences of points and nottoodegenerate hyperplanes in three or higherdimensional Euclidean spaces. We call a hyperplane nottoodegenerate if at most aconstant portion of its ..."
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Cited by 7 (3 self)
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We present a multidimensional generalization of the SzemerédiTrotter Theorem, and give asharp bound on the number of incidences of points and nottoodegenerate hyperplanes in three or higherdimensional Euclidean spaces. We call a hyperplane nottoodegenerate if at most aconstant portion of its incident points lie in a lower dimensional affine subspace.
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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Cited by 5 (0 self)
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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.
PointLine Incidences in Space
, 2002
"... Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints. ..."
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Cited by 5 (4 self)
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Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints.