# No-three-in-line problem

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.

## Lower bounds

Paul Erdős (in Template:Harvnb) 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

$(1-\epsilon )n$ points in the n × n grid with no three points collinear.

Erdős' bound has been improved subsequently: Template:Harvtxt show that, when n/2 is prime, one can obtain a solution with 3(n - 2)/2 points by placing points on the hyperbola xyk (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

$({\frac {3}{2}}-\epsilon )n$ points.

## Conjectured upper bounds

Template:Harvtxt conjectured that for large n one cannot do better than c n with

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

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

${\frac {1-\epsilon }{2n^{2}}}.$ ## 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 (Template:Harvtxt).

Non-collinear sets of points in the three-dimensional grid were considered by Template:Harvtxt. They proved that the maximum number of points in the n × n × n grid with no three points collinear is $\Theta (n^{2})$ . 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. 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 (Template:Harvtxt; Template:Harvtxt; Template:Harvtxt).

## 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 OEIS).