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. 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.

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 $t_3^\text{orchard}(n)$, 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 $n$ 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 $\tfrac{6n}{13}$ 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, 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, $n &gt; n_{0}$, 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. Thus, for sufficiently large $t_3^\text{orchard}(n)$, the exact value of $n$ is known.

This is slightly better than the bound that would directly follow from their tight lower bound of $n$ 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 $t_3^\text{orchard}(n)$ points lie in a projective plane defined over a finite field..