## No-Three-in-Line-in-3D
Pór, Attila and Wood, David R.
(2004)
Full text not available from this repository. ## AbstractThe 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 \Theta (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 \Theta(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 \mathbb{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 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 \Theta(n^3/2). This compares favourably to \Theta (n^3) when edges are not allowed to cross. Generalising the construction for K_n, we prove that every k-colourable graph on n vertices has a 3D drawing with O (n\sqrt{k}) volume. For the k-partite Turán graph, we prove a lower bound of \Omega((kn)^3/4). Research supported by NSERC and COMBSTRU.
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