Orchard-planting problem

An arrangement of nine points (related to the Pappus configuration) 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 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 (sequence A003035 in the OEIS).

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

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

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

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

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] 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.

Notes[]

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, S2CID 120906839.
• 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, JSTOR 2045427.
• Green, Ben; Tao, Terence (2013), "On sets defining few ordinary lines", Discrete and Computational Geometry, 50 (2): 409–468, arXiv:1208.4714, doi:10.1007/s00454-013-9518-9, S2CID 15813230

External links[]

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