Orchard-planting problem

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An arrangement of nine points forming ten 3-point lines.

In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of points in the plane. It is also called the tree-planting problem or simply the orchard problem. There are also investigations into how many k-point lines there can be. Hallard T. Croft and Paul Erdős proved tk > c n2 / k3, 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[edit]

Define t3orchard(n) to be the maximum number of 3-point lines attainable with a configuration of n points. For an arbitrary number of points, n, t3orchard(n) was shown to be (1/6)n2 − O(n) in 1974.

The first few values of t3orchard(n) are given in the following table (sequence A003035 in OEIS).

n 4 5 6 7 8 9 10 11 12 13 14
t3orchard(n) 1 2 4 6 7 10 12 16 19 22 26

Upper and lower bounds[edit]

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

\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 6n/13 (Csima & Sawyer 1993), this upper bound can be lowered to

\left\lfloor\frac{\binom{n}{2}-6n/13}{3}\right\rfloor=\left\lfloor\frac{n^2}{6}-\frac{25n}{78}\right\rfloor.

Lower bounds for t3orchard(n) are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ~(1/8)n2 was given by Sylvester, who placed n points on the cubic curve y = x3. This was improved to [(1/6)n2 − (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)n2 − (1/2)n + 1] 3-point lines which matches the lower bound established by Burr, Grünbaum and Sloane.[2] This is slightly better than the bound that would directly follow from their tight lower bound of n/2 for the number of 2-point lines: [n(n − 2)/6], proved in the same paper and solving a 1951 problem posed independently by Gabriel Andrew Dirac and Theodore Motzkin.

Notes[edit]

  1. ^ The Handbook of Combinatorics, edited by László Lovász, Ron Graham, et al, in the chapter entitled Extremal Problems in Combinatorial Geometry by Paul Erdős and George B. Purdy.
  2. ^ Green & Tao (2013)

References[edit]


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