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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 127 (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 17 (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.
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.