Orchard-planting problem

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

The first few values of ${\displaystyle t_{3}^{\text{orchard}}(n)}$ are given in the following table (sequence A003035 in the OEIS).

n	4	5	6	7	8	9	10	11	12	13	14
${\displaystyle t_{3}^{\text{orchard}}(n)}$	1	2	4	6	7	10	12	16	19	22	26

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 ${\displaystyle {\tfrac {6n}{13}}}$ (Csima & Sawyer 1993), this upper bound can be lowered to

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

Lower bounds for ${\displaystyle t_{3}^{\text{orchard}}(n)}$ are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ${\displaystyle \approx {\tfrac {1}{8}}n^{2}}$ was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to ${\displaystyle {\tfrac {1}{6}}n^{2}-{\tfrac {1}{2}}n+1}$ 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

$${\displaystyle {\frac {n(n-3)}{6}}+1={\frac {1}{6}}n^{2}-{\frac {1}{2}}n+1}$$

3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.[2] Thus, for sufficiently large n, the exact value of ${\displaystyle t_{3}^{\text{orchard}}(n)}$ is known.

This is slightly better than the bound that would directly follow from their tight lower bound of ${\displaystyle {\tfrac {n}{2}}}$ for the number of 2-point lines: ${\displaystyle {\tfrac {n(n-2)}{6}},}$ 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.(Padmanabhan & Shukla 2020).

• 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
• Padmanabhan, R.; Shukla, Alok (2020), "Orchards in elliptic curves over finite fields", Finite Fields and Their Applications, 68 (2): 101756, arXiv:2003.07172, doi:10.1016/j.ffa.2020.101756, S2CID 212725631

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