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A lower bound for Heilbronn's problem
 J. London Math. Soc
, 1982
"... We disprove Heilbronn's conjecture—that N points lying in the unit disc necessarily contain a triangle of area less than c/N 2. ..."
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We disprove Heilbronn's conjecture—that N points lying in the unit disc necessarily contain a triangle of area less than c/N 2.
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 32 (20 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
On Simultaneous Planar Graph Embeddings
 COMPUT. GEOM
, 2003
"... We consider the problem of simultaneous embedding of planar graphs. There are two variants ..."
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Cited by 29 (8 self)
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We consider the problem of simultaneous embedding of planar graphs. There are two variants
Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 26 (15 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
On Heilbronn's triangle problem
 J. London Math. Soc
, 1981
"... Let PuP2,...,Pn be a distribution of n points (where n ^ 3) in a closed convex region K of unit area such that the minimum of the areas of the triangles PjPjPk (taken over all selections of three out of n points) assumes its maximum possible value A(/C; n). We write A(n) = A(D; n) if D is a disc. ..."
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Cited by 23 (1 self)
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Let PuP2,...,Pn be a distribution of n points (where n ^ 3) in a closed convex region K of unit area such that the minimum of the areas of the triangles PjPjPk (taken over all selections of three out of n points) assumes its maximum possible value A(/C; n). We write A(n) = A(D; n) if D is a disc.
A Lower Bound for Heilbronn's Triangle Problem in d Dimensions
, 2001
"... . In this paper we show a lower bound for the generalization of Heilbronn's triangle problem to d dimensions; namely, we show that there exists a set S of n points in the ddimensional unit cube so that every d + 1 points of S define a simplex of volume## 1 n d ). We also show a constructive ..."
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. In this paper we show a lower bound for the generalization of Heilbronn's triangle problem to d dimensions; namely, we show that there exists a set S of n points in the ddimensional unit cube so that every d + 1 points of S define a simplex of volume## 1 n d ). We also show a constructive incremental positioning of n points in a unit 3cube for which every tetrahedron defined by four of these points has volume## 1 n 4 ). Key words. Heilbronn's triangle problem, probabilistic method AMS subject classifications. 05D40, 51M16 PII. S0895480100365859 1.
Collinear points in permutations
 Ann. Comb
"... Consider the following problem: how many collinear triples of points must a transversal of Zn×Zn have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n − 1)/4 and (n − 1)/2, and consider an analogous question for collinear qua ..."
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Cited by 6 (2 self)
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Consider the following problem: how many collinear triples of points must a transversal of Zn×Zn have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n − 1)/4 and (n − 1)/2, and consider an analogous question for collinear quadruples. We conjecture that the upper bound is the truth and suggest several other interesting problems in this area. In [4], Erdős offered a construction concerning the “Heilbronn Problem”. What is the smallest A so that, for any choice of n points in the unit square, some triangle formed by three of the points has area at most A? His elegant construction of a pointset with large minimumarea triangle ( ∼ n −2) is as follows: take the smallest prime p ≥ n, and let the set of points be {p −1 (x, x 2 (mod p)) : x ∈ Zp}. (If necessary, throw out a few points so that there are n left.) It is easy to see that this set has no three collinear points, and therefore any three points form a nondegenerate lattice triangle – which must have area at least p −2 /2 ≫ n −2. Another area in which collinear triples of points on a lattice arise is in connection with the socalled “nothreeinline ” problem, dating back at least to 1917 ([1]). Is it possible to choose 2n points on the nbyn grid so that no three are collinear? Clearly, if this is the case, then 2n is best possible. Guy and Kelly ([2]) conjecture that, for sufficiently large n, not only is it true that every set of 2n points has a collinear triple, but that it is possible to avoid collinear triples in a set of size (α −ǫ)n and impossible to avoid them in a set of size (α + ǫ)n, where α = (2π 2 /3) 1/3 ≈ 1.874 and ǫ> 0. In this note, we address the question of when it is possible to avoid collinear triples modulo n, particularly in the case of transversals (i.e., graphs of permutations) and when n is prime. 1 1
A Deterministic Polynomial Time Algorithm for Heilbronn’s Problem in Three Dimensions
 SIAM Journal on Computing
, 2002
"... Heilbronn conjectured that among arbitrary n points in the 2dimensional unit square [0, 1] 2, there must be three points which form a triangle of area at most O(1/n 2). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [14] who showed that for every n there ..."
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Cited by 5 (5 self)
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Heilbronn conjectured that among arbitrary n points in the 2dimensional unit square [0, 1] 2, there must be three points which form a triangle of area at most O(1/n 2). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [14] who showed that for every n there is a configuration of n points in the unit square [0, 1] 2 where all triangles have area at least Ω(log n/n 2). Here we will consider a 3dimensional analogue of this problem and we will give a deterministic polynomial time algorithm which finds n points in the unit cube [0, 1] 3 such that the volume of every tetrahedron among these n points is at least Ω(ln n/n 3). 1
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 4 (3 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