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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 107 (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.
On Distinct Sums and Distinct Distances
, 2001
"... The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bo ..."
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Cited by 13 (3 self)
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The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bound on the number of distinct sums. As an application we improve the SolymosiToth bound on an old Erd}os problem: the number of distinct distances n points determine in the plane. Our bound also nds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.
New Results on the Distribution of Distances Determined By Separated Point Sets
"... In this minisurvey, we summarize some joint results found by Paul Erd}os and the authors during the past decade on the set of distances between n points in Euclidean space. We concentrate on two types of questions: (1) How uniformly can the distances determined by such a point set be distribut ..."
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In this minisurvey, we summarize some joint results found by Paul Erd}os and the authors during the past decade on the set of distances between n points in Euclidean space. We concentrate on two types of questions: (1) How uniformly can the distances determined by such a point set be distributed? (2) What is the maximum number of distances that can lie in the union of k intervals of length 1, provided that the minimal distance is at least 1? Keywords and phrases: Interpoint distances in nite sets, uniform distribution, separated point sets, nearly equal distances 2000 Mathematics Subject Classi cation. 52C10 1