Orchard-planting problem

An arrangement of nine points forming ten 3-point lines.

In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration 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 (sequence 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 (Csima & Sawyer 1993), 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 Burr, Grünbaum, and Sloane (1974), using a construction based on Weierstrass's elliptic functions. An elementary construction using hypocycloids was found by Füredi & Palásti (1984) 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] 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.

Notes

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. Green & Tao (2013)

References

• Brass, P.; Moser, W. O. J.; Pach, J. (2005), Research Problems in Discrete Geometry, Springer-Verlag, ISBN 0-387-23815-8.
• Burr, S. A.; Grünbaum, B.; Sloane, N. J. A. (1974), "The Orchard problem", Geometriae Dedicata 2 (4): 397–424, doi:10.1007/BF00147569.
• Csima, J.; Sawyer, E. (1993), "There exist 6n/13 ordinary points", Discrete and Computational Geometry 9: 187–202, doi:10.1007/BF02189318.
• Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles", Proceedings of the American Mathematical Society 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427.
• Green, Ben; Tao, Terence (2013), "On sets defining few ordinary lines", Discrete and Computational Geometry 50 (2): 409–468, doi:10.1007/s00454-013-9518-9

External links

• Weisstein, Eric W., "Orchard-Planting Problem", MathWorld.