No-three-in-line problem

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A set of 20 points in a 10 × 10 grid, with no three points in a line.

In mathematics, in the area of discrete geometry, the no-three-in-line-problem, introduced by Henry Dudeney in 1917, asks for the maximum number of points that can be placed in the n × n grid so that no three points are collinear. This number is at most 2n, since if 2n + 1 points are placed in the grid some row will contain three points.

1 Lower bounds
2 Conjectured upper bounds
3 Applications
4 Generalizations
5 Small values of n
6 References
7 External links

[edit] Lower bounds

Paul Erdős (in Roth, 1951) observed that, when n is a prime number, the set of n grid points (i, i² mod n) contains no three collinear points. When n is not prime, one can perform this construction for a p × p grid contained in the n × n grid, where p is the largest prime that is at most n. As a consequence, for any ε and any sufficiently large n, one can place

(1 − ε)n

points in the n × n grid with no three points collinear.

Erdős' bound has been improved subsequently: Hall et al (1975) show that, when n/2 is prime, one can obtain a solution with 3(n - 2)/2 points by placing points on the hyperbola xy ≡ k (mod n/2) for a suitable k. Again, for arbitrary n one can perform this construction for a prime near n/2 to obtain a solution with

(3/2 − ε)n

points.

[edit] Conjectured upper bounds

Guy and Kelly (1968) conjectured that one cannot do better, for large n, than cn with

$c = \sqrt[3]{\frac{2\pi^2}{3}} \approx 1.874.$

In 2004, Guy refined this estimate, based on a communication by Gabor Ellmann, to

$c = \frac{\pi}{\sqrt 3} \approx 1.814.$

[edit] Applications

Heilbronn's problem asks for the placement of n points in a unit square that maximizes the area of the smallest triangle formed by three of the points. By applying Erdős' construction of a set of grid points with no three collinear points, one can find a placement in which the smallest triangle has area

$\frac{1-\epsilon}{2n^2}.$

[edit] Generalizations

A noncollinear placement of n points can also be interpreted as a graph drawing of the complete graph in such a way that, although edges cross, no edge passes through a vertex. Erdős' construction above can be generalized to show that every n-vertex k-colorable graph has such a drawing in a O(n) x O(k) grid (Wood 2005).

Non-collinear sets of points in the three-dimensional grid were considered by Pór and Wood (2007). They proved that the maximum number of points in the n x n x n grid with no three points collinear is $Θ(n 2)$ . One can also consider graph drawings in the three-dimensional grid. Here the non-collinearity condition means that a vertex should not lie on a non-adjacent edge, but it is normal to work with the stronger requirement that no two edges cross (Pach et al 1998; Dujmović et al 2005; Di Giacomo 2005).

[edit] Small values of n

For n ≤ 32, it is known that 2n points may be placed with no three in a line. The numbers of solutions (not counting reflections and rotations as distinct) for small n = 1, 2, ..., are

1, 1, 4, 5, 11, 22, 57, 51, 156, 158, 566, 499, 1366, ... (sequence A000769 in OEIS).

[edit] References

Dudeney, Henry (1917). Amusements in Mathematics. Edinburgh: Nelson.

Emilio Di Giacomo, Giuseppe Liotta, and Henk Meijer (2005). "Computing Straight-line 3D Grid Drawings of Graphs in Linear Volume". Comput. Geom. 32 (1): 26-58.

Vida Dujmović, Pat Morin, and David R. Wood (2005). "Layout of Graphs with Bounded Tree-Width". SIAM J. Comput. 34 (3): 553-579.

Stefan Felsner, Giussepe Liotta, and Stephen K. Wismath (2003). "Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions". J. Graph Algorithms & Applications 7 (4): 363-398.

Flammenkamp, Achim (1992). "Progress in the no-three-in-line problem". Journal of Combinatorial Theory, Ser. A 60 (2): 305–311.

Flammenkamp, Achim (1998). "Progress in the no-three-in-line problem, II". Journal of Combinatorial Theory, Ser. A 81 (1): 108–113.

Guy, R. K.; Kelly, P. A. (1968). "The no-three-in-line problem". Canad. Math. Bull. 11: 527–531.

Hall, R. R.; Jackson, T. H.; Sudbery, A.; Wild, K. (1975). "Some advances in the no-three-in-line problem". Journal of Combinatorial Theory, Ser. A 18: 336–341.

Pach, János; Thiele, Torsten; Tóth, Géza (1998). "Three-dimensional grid drawings of graphs". Proc. 5th Int. Worksh. Graph Drawing (GD '97): 47–51, Lecture Notes in Computer Science, no. 1353, Springer-Verlag.