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 > c n 2 k 3 , {\displaystyle t_{k}>{\frac {cn^{2}}{k^{3}}},} where n is the number of points and tk 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 t 3 orchard ( n ) {\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, t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} was shown to be 1 6 n 2 − O ( n ) {\displaystyle {\tfrac {1}{6}}n^{2}-O(n)} in 1974. The first few values of t 3 orchard ( n ) {\displaystyle t_{3}^{\text{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
⌊ ( n 2 ) / ( 3 2 ) ⌋ = ⌊ n 2 − n 6 ⌋ . {\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 6 n 13 {\displaystyle {\tfrac {6n}{13}}} (Csima & Sawyer 1993), this upper bound can be lowered to
⌊ ( n 2 ) − 6 n 13 3 ⌋ = ⌊ n 2 6 − 25 n 78 ⌋ . {\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 t 3 orchard ( n ) {\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 ≈ 1 8 n 2 {\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 1 6 n 2 − 1 2 n + 1 {\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
n ( n − 3 ) 6 + 1 = 1 6 n 2 − 1 2 n + 1 {\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. Thus, for sufficiently large n, the exact value of t 3 orchard ( n ) {\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 n 2 {\displaystyle {\tfrac {n}{2}}} for the number of 2-point lines: n ( n − 2 ) 6 , {\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).
See also Arrangement of lines
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 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