No-three-in-line problem

In mathematics, in the area of discrete geometry, the no-three-in-line problem 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, then by the pigeonhole principle some row and some column will contain three points. The problem was introduced by Henry Dudeney in 1917.

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

Although the problem can be solved with 2n points for every n up to 46, it is conjectured that fewer than 2n points are possible for sufficiently large values of n. The best solutions that are known to work for arbitrarily large values of n place slightly fewer than 3n/2 points.

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), for 0 ≤ i < 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

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), where k may be chosen arbitrarily as long as it is nonzero mod n/2. Again, for arbitrary n one can perform this construction for a prime near n/2 to obtain a solution with

points.

Conjectured upper bounds

Guy & Kelly (1968) conjectured that for large n one cannot do better than c n with

Pegg (2005) noted that Gabor Ellmann found, in March 2004, an error in the original paper of Guy and Kelly's heuristic reasoning, which if corrected, results in

Applications

The Heilbronn triangle 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

Generalizations

Higher dimensions

Non-collinear sets of points in the three-dimensional grid were considered by Pór & Wood (2007). They proved that the maximum number of points in the n × n × n grid with no three points collinear is . Similarly to Erdős's 2D construction, this can be accomplished by using points (x, y, x2 + y2) mod p, where p is a prime congruent to 3 mod 4.

Another analogue in higher dimensions is to find sets of points that do not all lie in the same plane (or hyperplane). For the no-four-coplanar problem in three dimensions, it was reported by Ed Pegg, Oleg567 et al, that 8 such points can be selected in a 3x3x3 grid (exactly one solution up to rotation/reflection), 10 such points can be found for 4x4x4 (232 different solutions), and 13 such points can be found for 5x5x5 (38 different solutions).[1][2] As of 2015, it is not known what the maximal solution is for 6x6x6 grid, nor how many such solutions exist. Similar to the 2n upper bound for the 2D case, there exists a 3n upper bound for the 3D case (no more than 3 points per plane, and no more than n such planes in the grid), though as seen above, not all values of n attain the upper bound.

The cap set problem concerns a similar problem in high-dimensional vector spaces over finite fields.[3]

Graph 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) × O(k) grid (Wood 2005).

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, Thiele & Tóth 1998; Dujmović, Morin & Wood 2005; Di Giacomo, Liotta & Meijer 2005).

Small values of n

For n ≤ 46, 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 = 2, 3, ..., are

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

Notes
  1. Ed Pegg's question
  2. Ed Pegg's site
  3. Klarreich, Erica (May 31, 2016), "Simple Set Game Proof Stuns Mathematicians", Quanta.

References
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