The orchard problem

Burr, Stefan A.; Grünbaum, Branko; Sloane, N. J. A.

doi:10.1007/bf00147569

Cited by 50 publications

(55 citation statements)

References 27 publications

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“…This theorem is tight for large n, as noted by Sylvester [34], and also subsequently by Burr, Grünbaum and Sloane [5], who discuss this problem extensively. We will give these examples, which are based on irreducible cubic curves, in Proposition 2.6 below.…”

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 61%

“…5. This quickly implies that P is close, up to projective transformation, to one of the known extremisers coming from roots of unity or from subgroups of elliptic curves, with a small number of additional points added or removed.…”

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

“…As observed in [5], lower bounds for the number N 2 of ordinary lines can be converted into upper bounds for the number N 3 of 3-rich lines thanks to the obvious double-counting identity n k=2 k 2 N k = n 2 (with N k denoting the number of k-rich lines). In particular, previously known lower bounds on the Dirac-Motzkin conjecture can be used to deduce upper bounds on the orchard problem.…”

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

“…This allows us to rule out "essentially torsion-free" situations, such as that provided by singular cubic curves (except for the acnodal singular cubic curve), and eventually leads to the full structure theorem in Theorem 1. 5.…”

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

“…We turn now to Sylvester's examples of point sets coming from cubic curves, as further discussed by Burr, Grünbaum and Sloane [5]. While these do not provide the best examples of sets with few ordinary lines, it appears that consideration of them is essential in order to solve the Dirac-Motzkin problem.…”

Section: Theorem 24 (Strong Dirac-motzkin Conjecture)mentioning

confidence: 99%

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On Sets Defining Few Ordinary Lines

Green¹,

Tao²

2013

Discrete Comput Geom

167

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Abstract. Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P . This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n − C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P . Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

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Section: Theorem 13 (Orchard Problem)mentioning

confidence: 61%

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

Section: Theorem 13 (Orchard Problem)mentioning

confidence: 99%

Section: Theorem 24 (Strong Dirac-motzkin Conjecture)mentioning

confidence: 99%

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On Sets Defining Few Ordinary Lines

Green¹,

Tao²

2013

Discrete Comput Geom

167

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The Maximum Number of Triangles in Arrangements of Pseudolines

Roudneff

1996

Journal of Combinatorial Theory, Series B

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Gru nbaum has conjectured that any arrangement of n pseudolines in the real projective plane has at most 1 3 n(n&1) triangular faces when n is sufficiently large. We prove this conjecture for n 9; the result does not hold for n 8. The structure of extremal examples is explored and an infinite family of non simple arrangements with 1 3 n(n&1) triangles is constructed. As an application, we show that the number of simplices in arrangements of n 10 pseudoplanes is always less than 1 12 n(n&1)(n&2).

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Planting Trees

Burr

1981

The Mathematical Gardner

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The orchard problem

Cited by 50 publications

References 27 publications

On Sets Defining Few Ordinary Lines

On Sets Defining Few Ordinary Lines

The Maximum Number of Triangles in Arrangements of Pseudolines

Planting Trees

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