Orchard-planting problem

Not to be confused with Euclid's orchard.

File:Orchard-planting problem.svg

An arrangement of nine points (related to the Pappus configuration) forming ten 3-point lines.

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 $$t_k>\frac{c{n}^2}{k^3},$$ 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

File:Orchard planting problem.svg

Maximum possible number of 3-point lines 𝑡₃ for 𝑛 from 3 to 11.

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 $\frac{1}{6}{n}^2-O(n)$ in 1974.

The first few values of ⁠$t_3^{\text{orchard}}(n)$⁠ are given in the following table (sequence A003035 in the OEIS).

n	4	5	6	7	8	9	10	11	12	13	14
⁠$t_3^{\text{orchard}}(n)$⁠	1	2	4	6	7	10	12	16	19	22	26

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 $$\lfloor (\genfrac{}{}{0ex}{}{n}{2})/(\genfrac{}{}{0ex}{}{3}{2})\rfloor =\lfloor \frac{n^2-n}{6}\rfloor .$$ Using the fact that the number of 2-point lines is at least ⁠$\frac{6n}{13}$⁠ (Csima & Sawyer 1993), this upper bound can be lowered to $$\lfloor \frac{(\genfrac{}{}{0ex}{}{n}{2})-\frac{6n}{13}}{3}\rfloor =\lfloor \frac{n^2}{6}-\frac{25n}{78}\rfloor .$$

Lower bounds for ⁠$t_3^{\text{orchard}}(n)$⁠ are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of $\approx \frac{1}{8}{n}^2$ was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to $\frac{1}{6}{n}^2-\frac{1}{2}n+1$ 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 $$\frac{n(n-3)}{6}+1=\frac{1}{6}{n}^2-\frac{1}{2}n+1$$ 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane. Thus, for sufficiently large n, the exact value of ⁠$t_3^{\text{orchard}}(n)$⁠ is known.

This is slightly better than the bound that would directly follow from their tight lower bound of ⁠$\frac{n}{2}$⁠ for the number of 2-point lines: $\frac{n(n-2)}{6},$ 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

References

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