Since no two lines may share two distinct points, a
trivial upper-bound for the number of 3-point lines determined by 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 {{tmath|\tfrac{6n}{13} }} , 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 {{tmath|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 points on the cubic curve . This was improved to \tfrac{1}{6}n^2 - \tfrac{1}{2}n + 1 in 1974 by , using a construction based on
Weierstrass's elliptic functions. An elementary construction using
hypocycloids was found by 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, , 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. Thus, for sufficiently large , the exact value of {{tmath|t_3^\text{orchard}(n)}} is known. This is slightly better than the bound that would directly follow from their tight lower bound of 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 points lie in a projective plane defined over a finite field. . ==See also==