Results 1  10
of
20
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 ..."
Abstract

Cited by 158 (1 self)
 Add to MetaCart
(Show Context)
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.
SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS, AND THE DIMENSION OF EXCEPTIONS
 VOL. 102, NO. 2 DUKE MATHEMATICAL JOURNAL
, 2000
"... ..."
Some connections between Falconer’s distance set conjecture, and sets of Furstenburg type
, 2001
"... ..."
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 ..."
Abstract

Cited by 27 (8 self)
 Add to MetaCart
(Show Context)
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.
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. ..."
Abstract

Cited by 21 (0 self)
 Add to MetaCart
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.
The number of distinct distances from a vertex of a convex polygon
 J. Comput. Geom
"... Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argume ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argument is showing that P must determine at most n2 (1 − 1/12) isosceles triangles. In this paper we show that Dumitrescu’s bound can be further improved, though our improvement is quite small. We show that the number of isosceles triangles determined by P is at most n2 (1 − 1/11.981), and we conclude that there exists a point of P that determines at least
The Beginnings of Geometric Graph Theory
"... “...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
“...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erdős from 1946, we give a biased survey of Turántype questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erdős or directly motivated by his work. 1
The Exact Fitting . . .
 COMPUT. GEOM. THEORY APPL
, 1992
"... Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This a ..."
Abstract
 Add to MetaCart
Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This algorithm is based on upper bounds on the maximum number of incidences between families of points and families ofhyperplanes in E d and on an algorithm to compute these incidences. We also show how the upper bound on the maximum number of incidences between families of points and families of hyperplanes can be used to derive new bounds on some wellknown problems in discrete geometry.