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21
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.
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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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.
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 ..."
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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.
ON SOME METRIC AND COMBINATORIAL GEOMETRIC PROBLEMS
, 1986
"... Let x x „ be n distinct points in the plane. Denote by D(x xn) the minimum number of distinct distances determined by x x,,. Put f (n) = min D(x xX"). An old and probably very difficult conjecture of mine states that fin)> cn/(log n)l. f(5)=2 and the only way we can get f(5)=2 is if the poi ..."
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Cited by 11 (0 self)
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Let x x „ be n distinct points in the plane. Denote by D(x xn) the minimum number of distinct distances determined by x x,,. Put f (n) = min D(x xX"). An old and probably very difficult conjecture of mine states that fin)> cn/(log n)l. f(5)=2 and the only way we can get f(5)=2 is if the points form a regular pentagon. Are there other values of n for which there is a unique configuration of points for which the minimal value of f(n) is assumed? Is it true that the set of points which implements f(n) has lattice structure? Many related questions are discussed.
private communication
, 1983
"... The ndimensional lattices that contain fewest distances are characterized for all n # 2. 0 1991 Academic Press, Inc. 1. ..."
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The ndimensional lattices that contain fewest distances are characterized for all n # 2. 0 1991 Academic Press, Inc. 1.
Cardinalities of kdistance sets in Minkowski spaces
 Discrete Mathematics
, 1999
"... Abstract. 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) d points, with equality iff the unit ball is a parallelotope. ..."
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Abstract. 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) d points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we find various weaker upper bounds for kdistance sets. 1.
Lattices with Few Distances
 J. Number Theory
, 1991
"... this paper we prove all cases except n = 2 (for which see Smith [37]) of the following proposition. ..."
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this paper we prove all cases except n = 2 (for which see Smith [37]) of the following proposition.
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 ..."
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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 ..."
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“...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