Orchard-planting problem

Short description: Geometry; how many 3-point lines can n points form

An arrangement of nine points (related to the Pappus configuration) forming ten 3-point lines.

In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. There are also investigations into how many k-point lines there can be. Hallard T. Croft and Paul Erdős proved [math]\displaystyle{ t_k \gt \frac{cn^2}{k^3}, }[/math] where n is the number of points and t_k is the number of k-point lines. Their construction contains some m-point lines, where m > k. One can also ask the question if these are not allowed.

Integer sequence

Define [math]\displaystyle{ t_3^\text{orchard}(n) }[/math] to be the maximum number of 3-point lines attainable with a configuration of n points. For an arbitrary number of n points, [math]\displaystyle{ t_3^\text{orchard}(n) }[/math] was shown to be [math]\displaystyle{ \tfrac{1}{6}n^2 - O(n) }[/math] in 1974.

The first few values of [math]\displaystyle{ t_3^\text{orchard}(n) }[/math] are given in the following table (sequence A003035 in the OEIS).

n	4	5	6	7	8	9	10	11	12	13	14
[math]\displaystyle{ t_3^\text{orchard}(n) }[/math]	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 [math]\displaystyle{ \left\lfloor \binom{n}{2} \Big/ \binom{3}{2} \right\rfloor = \left\lfloor \frac{n^2-n}{6} \right\rfloor. }[/math] Using the fact that the number of 2-point lines is at least [math]\displaystyle{ \tfrac{6n}{13} }[/math] (Csima Sawyer), this upper bound can be lowered to [math]\displaystyle{ \left\lfloor \frac{\binom{n}{2} - \frac{6n}{13}}{3} \right\rfloor = \left\lfloor \frac{n^2}{6} - \frac{25n}{78} \right\rfloor. }[/math]

Lower bounds for [math]\displaystyle{ t_3^\text{orchard}(n) }[/math] are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of [math]\displaystyle{ \approx \tfrac{1}{8}n^2 }[/math] was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to [math]\displaystyle{ \tfrac{1}{6}n^2 - \tfrac{1}{2}n + 1 }[/math] 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) 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 [math]\displaystyle{ \frac{n(n-3)}{6} + 1 = \frac{1}{6}n^2 - \frac{1}{2}n + 1 }[/math] 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane. Thus, for sufficiently large n, the exact value of [math]\displaystyle{ t_3^\text{orchard}(n) }[/math] is known.

This is slightly better than the bound that would directly follow from their tight lower bound of [math]\displaystyle{ \tfrac n 2 }[/math] for the number of 2-point lines: [math]\displaystyle{ \tfrac{n(n-2)}{6}, }[/math] 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.(Padmanabhan Shukla).

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 (2): 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 .
• 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 
• Padmanabhan, R.; Shukla, Alok (2020), "Orchards in elliptic curves over finite fields", Finite Fields and Their Applications 68 (2): 101756, doi:10.1016/j.ffa.2020.101756 

External links

• Weisstein, Eric W.. "Orchard-Planting Problem". http://mathworld.wolfram.com/Orchard-PlantingProblem.html.