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11
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.
Survey of the Steinhaus tiling problem
, 2003
"... We survey some results and problems arising from a classic problem of Steinhaus: Is there a subset S of R² such that each isometric copy of Z² (the lattice points in the plane) meets S in exactly one point. ..."
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Cited by 7 (1 self)
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We survey some results and problems arising from a classic problem of Steinhaus: Is there a subset S of R² such that each isometric copy of Z² (the lattice points in the plane) meets S in exactly one point.
The averagecase area of Heilbronntype triangles
 RANDOM STRUCTURES AND ALGORITHMS
, 2002
"... From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points ..."
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Cited by 6 (2 self)
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From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points are chosen independently and at random (with a uniform distribution), then there exist positive constants c and C such that c/n3 <µ n < C/n3 for all large enough values of n, where µ n is the expectation of A. Moreover, c/n3 <A<C/n3, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in
The expected size of Heilbronn's triangles
 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
, 1999
"... Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C su ..."
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Cited by 5 (4 self)
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Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n 2 ≤ ∆ ≤ C/n 8/7−ǫ for every constant ǫ> 0. We resolve Heilbronn’s problem in the expected case: If we uniformly at random put n points in the unit disc then (i) the area of the smallest triangle has expectation Θ(1/n 3); and (ii) the smallest triangle has area Θ(1/n 3) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity. 1
Restricted Point Configurations with Many Collinear ktuplets
, 2001
"... Given k 3, denote by t 0 k (N) the largest integer for which there is a set of N points in the plane, no k + 1 of them on a line such that there are t 0 k (N) lines, each containing exactly k of the points. Erd}os (1962) raised the problem of estimating the order of magnitude of t 0 k (N ). ..."
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Cited by 5 (0 self)
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Given k 3, denote by t 0 k (N) the largest integer for which there is a set of N points in the plane, no k + 1 of them on a line such that there are t 0 k (N) lines, each containing exactly k of the points. Erd}os (1962) raised the problem of estimating the order of magnitude of t 0 k (N ). We prove that: t 0 k (N) c 0 k N log(k+4)=log(k) improving a previous bound of Grunbaum for all 5 k 35. 1
SOME OLD AND NEW PROBLEMS IN COMBINATORIAL GEOMETRY
, 1984
"... In this paper several unconnected old and new problems in combinatorial geometry are discussed. The reference list is as complete as possible, including two papers dealing with similar subjects [6, 7]. Many interesting problems and results are in Grünbaum [11], which also has an extensive and useful ..."
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In this paper several unconnected old and new problems in combinatorial geometry are discussed. The reference list is as complete as possible, including two papers dealing with similar subjects [6, 7]. Many interesting problems and results are in Grünbaum [11], which also has an extensive and useful bibliography and interesting historical remarks.
On a Lattice Problem of H. Steinhaus
"... . It is shown that there is a subset S of R 2 such that each isometric copy of Z 2 (the lattice points in the plane) meets S in exactly one point. This provides a positive answer to a problem of H. Steinhaus. 1. Introduction Sometime in the 1950's, Steinhaus posed the following problem. Do ther ..."
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. It is shown that there is a subset S of R 2 such that each isometric copy of Z 2 (the lattice points in the plane) meets S in exactly one point. This provides a positive answer to a problem of H. Steinhaus. 1. Introduction Sometime in the 1950's, Steinhaus posed the following problem. Do there exist two sets A and B in the plane such that every set congruent to A has exactly one point in common with B? The trivial case where one of the sets is the plane and the other consists of a single point is ruled out. The rst appearance of this problem in the literature seems to be in a 1958 paper of Sierpinski [11]. In this paper, he showed the answer is yes, a result later rediscovered by Erd}os [5]. Of course, there are many variants of this problem. For example, one could specify the set A. In this direction, Komjath showed that such a set exists if A = Z, the set of all integers [10]. Steinhaus also asked about the specic case where A = Z 2 . The rst reference to this problem a...
Kolmogorov Complexity and a Triangle Problem of the Heilbronn Type
"... From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such tha ..."
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From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c=n 3 ! n ! C=n 3 for all large enough n, where n is the expectation of A. Moreover, with probability close to one c=n 3 ! A ! C=n 3 . Our proof uses the incompressibility method based on Kolmogorov complexity. The related Heilbronn problem asks for the maximum value assumed by A over all choices of n points. 1 Introduction From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit circle, let T be the one of least area, and let A be the area of T . Let \Delta n be the maximum assumed by A over all choices of n points. H.A. Heilbronn (19081975) 1 asked for the exact value or approximation of \Delta n . The list [1, 2, 3, 5, 8, 9, 10, 11,...