Benutzer:Paul Hege/Obstgartenproblem

Als Obstgarten-Problem (englisch orchard problem) wird in der Mathematik die folgende Fragestellung bezeichnet: Wie können neun Obstbäume so angeordnet werden, dass sich zehn Sichtachsen mit jeweils drei Bäumen ergeben? Das allgemeinere Problem, wie $n$ Punkte in der Ebene so angeordnet werden können, dass möglichst viele Geraden mit jeweils $k$ Punkten darauf existieren, beschäftigt Mathematiker bis heute.

Obstgarten-Folge[Bearbeiten | Quelltext bearbeiten]

Sei $t_{3}(n)$ die maximale Anzahl von Geraden mit drei Punkten, die in einer Konfiguration von $n$ Punkten in der Ebene existieren können. Die ersten Glieder dieser Folge sind:

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

Diese Folge wird als Folge A003035 in OEIS aufgeführt.

Upper and lower bounds[Bearbeiten | Quelltext bearbeiten]

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 Vorlage:Harv, 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 t₃^orchard(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 Vorlage:Harvs, using a construction based on Weierstrass's elliptic functions. An elementary construction using hypocycloids was found by Vorlage:Harvtxt 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.^[1] Thus, for sufficiently large n, the exact value of t₃^orchard(n) is known.

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

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.Vorlage:Harv.