Orchard-planting problem

Define ⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠ to be the maximum number of 3-point lines attainable with a configuration of n points. For an arbitrary number of n points, ⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠ was shown to be ${\displaystyle {\tfrac {1}{6}}n^{2}-O(n)}$ in 1974.

The first few values of ⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠ are given in the following table (sequence A003035 in the OEIS).

More information ⁠ ...

n	4	5	6	7	8	9	10	11	12	13	14
⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠	1	2	4	6	7	10	12	16	19	22	26

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

Lower bounds for ⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠ are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ${\displaystyle \approx {\tfrac {1}{8}}n^{2}}$ was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to ${\displaystyle {\tfrac {1}{6}}n^{2}-{\tfrac {1}{2}}n+1}$ 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 $${\displaystyle {\frac {n(n-3)}{6}}+1={\frac {1}{6}}n^{2}-{\frac {1}{2}}n+1}$$ 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.^[4] Thus, for sufficiently large n, the exact value of ⁠${\displaystyle t_{3}^{\text{orchard}}(n)}$⁠ is known.

This is slightly better than the bound that would directly follow from their tight lower bound of ⁠${\displaystyle {\tfrac {n}{2}}}$⁠ for the number of 2-point lines: ${\displaystyle {\tfrac {n(n-2)}{6}},}$ 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 n points lie in a projective plane defined over a finite field. (Padmanabhan & Shukla 2020).

• Arrangement of lines