No-three-in-line-in-3D

The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n × n grid with no three points collinear. In 1951, Erdös proved that the answer is θ(n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n × n × n grid with no three collinear is θ(n ²). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in ℤ³, such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph K_n is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of K_n is θ(n ^3/2). This compares favourably to θ(n³) when edges are not allowed to cross. Generalising the construction for Kⁿ, we prove that every k-colourable graph on n vertices has a 3D drawing with O(n√k) volume. For the k-partite Turán graph, we prove a lower bound of Ω((kn)^3/4).