Definitions

# 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 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

(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 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

(3/2 − ε)n
points.

## Conjectured upper bounds

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

$c = sqrt\left[3\right]\left\{frac\left\{2pi^2\right\}\left\{3\right\}\right\} approx 1.874.$
In 2004, Guy refined this estimate, based on a communication by Gabor Ellmann, to
$c = frac\left\{pi\right\}\left\{sqrt 3\right\} approx 1.814.$

## 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

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

## 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 $Theta\left(n^2\right)$. 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).

## 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 = 2, 3, ..., are

1, 1, 4, 5, 11, 22, 57, 51, 156, 158, 566, 499, 1366, ... .

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