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
where n is the number of points and tk 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.
to be the maximum number of 3-point lines attainable with a configuration of n points.
For an arbitrary number of n points,
was shown to be
The first few values of
are given in the following table (sequence A003035 in the OEIS).
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
Using the fact that the number of 2-point lines is at least
(Csima & Sawyer 1993), this upper bound can be lowered to
Lower bounds for
are given by constructions for sets of points with many 3-point lines.
The earliest quadratic lower bound of
was given by Sylvester, who placed n points on the cubic curve y = x3.
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
3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.
[2] Thus, for sufficiently large n, the exact value of
This is slightly better than the bound that would directly follow from their tight lower bound of
for the number of 2-point lines:
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).
