No-three-in-line-in-3d

Abstract. 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(n2). This result is generalised by the notion of a 3D draw-ing of a graph. Here each vertex is represented by a distinct gridpoint in Z 3, 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 Kn 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 Kn 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).