Orchard-planting problem

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

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

$n$	4	5	6	7	8	9	10	11	12	13	14
⁠ $t_{3}^{\text{orchard}}(n)$ ⁠	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 ⁠ ${\tfrac {6n}{13}}$ ⁠ (Csima & Sawyer 1993), this upper bound can be lowered to $\left\lfloor {\frac {{\binom {n}{2}}-{\frac {6n}{13}}}{3}}\right\rfloor =\left\lfloor {\frac {n^{2}}{6}}-{\frac {25n}{78}}\right\rfloor .$

Lower bounds for ⁠ $t_{3}^{\text{orchard}}(n)$ ⁠ are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of $\approx {\tfrac {1}{8}}n^{2}$ was given by Sylvester, who placed $n$ points on the cubic curve $y = x 3$ . This was improved to ${\tfrac {1}{6}}n^{2}-{\tfrac {1}{2}}n+1$ 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, $n > n 0$ , there are at most ${\frac {n(n-3)}{6}}+1={\frac {1}{6}}n^{2}-{\frac {1}{2}}n+1$ 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.^[2] Thus, for sufficiently large $n$ , the exact value of ⁠ $t_{3}^{\text{orchard}}(n)$ ⁠ is known.

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

Notes

^ 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.
^ 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, 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

External links

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

[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)

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