Orchard-planting problem

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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 > c n 2 k 3 , {\displaystyle t_{k}>{\frac {cn^{2}}{k^{3}}},} {\displaystyle t_{k}>{\frac {cn^{2}}{k^{3}}},} 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.

Integer sequence

Maximum possible number of 3-point lines 𝑡3 for 𝑛 from 3 to 11.[2][3]

Define ⁠ t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\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, ⁠ t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\displaystyle t_{3}^{\text{orchard}}(n)}⁠ was shown to be 1 6 n 2 − O ( n ) {\displaystyle {\tfrac {1}{6}}n^{2}-O(n)} {\displaystyle {\tfrac {1}{6}}n^{2}-O(n)} in 1974.

The first few values of ⁠ t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\displaystyle 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 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\displaystyle 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 ⌊ ( n 2 ) / ( 3 2 ) ⌋ = ⌊ n 2 − n 6 ⌋ . {\displaystyle \left\lfloor {\binom {n}{2}}{\Big /}{\binom {3}{2}}\right\rfloor =\left\lfloor {\frac {n^{2}-n}{6}}\right\rfloor .} {\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 ⁠ 6 n 13 {\displaystyle {\tfrac {6n}{13}}} {\displaystyle {\tfrac {6n}{13}}}⁠ (Csima & Sawyer 1993), this upper bound can be lowered to ⌊ ( n 2 ) − 6 n 13 3 ⌋ = ⌊ n 2 6 − 25 n 78 ⌋ . {\displaystyle \left\lfloor {\frac {{\binom {n}{2}}-{\frac {6n}{13}}}{3}}\right\rfloor =\left\lfloor {\frac {n^{2}}{6}}-{\frac {25n}{78}}\right\rfloor .} {\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 ⁠ t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\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 ≈ 1 8 n 2 {\displaystyle \approx {\tfrac {1}{8}}n^{2}} {\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 1 6 n 2 − 1 2 n + 1 {\displaystyle {\tfrac {1}{6}}n^{2}-{\tfrac {1}{2}}n+1} {\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 n ( n − 3 ) 6 + 1 = 1 6 n 2 − 1 2 n + 1 {\displaystyle {\frac {n(n-3)}{6}}+1={\frac {1}{6}}n^{2}-{\frac {1}{2}}n+1} {\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 ⁠ t 3 orchard ( n ) {\displaystyle t_{3}^{\text{orchard}}(n)} {\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 ⁠ n 2 {\displaystyle {\tfrac {n}{2}}} {\displaystyle {\tfrac {n}{2}}}⁠ for the number of 2-point lines: n ( n − 2 ) 6 , {\displaystyle {\tfrac {n(n-2)}{6}},} {\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).

See also

Notes

References