Benutzer:Paul Hege/Obstgarten-Problem

In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard problem. There are also investigations into how many k-point lines there can be. Hallard T. Croft and Paul Erdős proved t_k > c n² / k³, where n is the number of points and t_k is the number of k-point lines.^[1] Their construction contains some m-point lines, where m > k. One can also ask the question if these are not allowed.

Integer sequence

Define t₃^orchard(n) to be the maximum number of 3-point lines attainable with a configuration of n points. For an arbitrary number of points, n, t₃^orchard(n) was shown to be (1/6)n² − O(n) in 1974.

The first few values of t₃^orchard(n) are given in the following table Folge A003035 in OEIS.

n	4	5	6	7	8	9	10	11	12	13	14
t₃^orchard(n)	1	2	4	6	7	10	12	16	19	22	26

Upper and lower bounds

Since no two lines may share two distinct points, a trivial upper-bound for the number of 3-point lines determined by n points is

\left\lfloor {\binom {n}{2}}{\Big /}{\binom {3}{2}}\right\rfloor =\left\lfloor {\frac {n^{2}-n}{6}}\right\rfloor .

Using the fact that the number of 2-point lines is at least 6n/13 Vorlage:Harv, this upper bound can be lowered to

\left\lfloor {\frac {{\binom {n}{2}}-6n/13}{3}}\right\rfloor =\left\lfloor {\frac {n^{2}}{6}}-{\frac {25n}{78}}\right\rfloor .

Lower bounds for t₃^orchard(n) are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ~(1/8)n² was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to [(1/6)n² − (1/2)n] + 1 in 1974 by Vorlage:Harvs, using a construction based on Weierstrass's elliptic functions. An elementary construction using hypocycloids was found by Vorlage:Harvtxt achieving the same lower bound.

In September 2013, Ben Green and Terence Tao published a paper in which they prove that for all point sets of sufficient size, n > n₀, there are at most ([n(n - 3)/6] + 1) = [(1/6)n² − (1/2)n + 1] 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.^[2] Thus, for sufficiently large n, the exact value of t₃^orchard(n) is known.

This is slightly better than the bound that would directly follow from their tight lower bound of n/2 for the number of 2-point lines: [n(n − 2)/6], proved in the same paper and solving a 1951 problem posed independently by Gabriel Andrew Dirac and Theodore Motzkin

Orchard-planting problem has also been considered over finite fields. In this version of the problem, the n points lie in a projective plane defined over a finite field.Vorlage:Harv.

Notes

Vorlage:Reflist

References

↑ The Handbook of Combinatorics, edited by László Lovász, Ron Graham, et al, in the chapter titled Extremal Problems in Combinatorial Geometry by Paul Erdős and George B. Purdy.
↑ Vorlage:Harvtxt

[1] The Handbook of Combinatorics, edited by László Lovász, Ron Graham, et al, in the chapter titled Extremal Problems in Combinatorial Geometry by Paul Erdős and George B. Purdy.

[2] Vorlage:Harvtxt

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