- Source: Orchard-planting problem
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}}},}
where n is the number of points and tk 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
Define
t
3
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)}
was shown to be
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)}
are given in the following table (sequence A003035 in the OEIS).
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 .}
Using the fact that the number of 2-point lines is at least
6
n
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 .}
Lower bounds for
t
3
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}}
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}
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}
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
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}}}
for the number of 2-point lines:
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).
Notes
References
Brass, P.; Moser, W. O. J.; Pach, J. (2005), Research Problems in Discrete Geometry, Springer-Verlag, ISBN 0-387-23815-8.
Burr, S. A.; Grünbaum, B.; Sloane, N. J. A. (1974), "The Orchard problem", Geometriae Dedicata, 2 (4): 397–424, doi:10.1007/BF00147569, S2CID 120906839.
Csima, J.; Sawyer, E. (1993), "There exist 6n/13 ordinary points", Discrete and Computational Geometry, 9 (2): 187–202, doi:10.1007/BF02189318.
Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles", Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427.
Green, Ben; Tao, Terence (2013), "On sets defining few ordinary lines", Discrete and Computational Geometry, 50 (2): 409–468, arXiv:1208.4714, doi:10.1007/s00454-013-9518-9, S2CID 15813230
Padmanabhan, R.; Shukla, Alok (2020), "Orchards in elliptic curves over finite fields", Finite Fields and Their Applications, 68 (2): 101756, arXiv:2003.07172, doi:10.1016/j.ffa.2020.101756, S2CID 212725631
External links
Weisstein, Eric W., "Orchard-Planting Problem", MathWorld
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