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
Define 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 in 1974.
The first few values of are given in the following table (sequence A003035 in the OEIS).
n | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 4 | 6 | 7 | 10 | 12 | 16 | 19 | 22 | 26 |
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
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. This was improved to 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
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).