No-Three-in-Line-in-3D

Abstract

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(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 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(n 3 ) 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 ).

Zero-Sum Problems in Finite Abelian Groups and Affine Caps

“…The interest is in drawings with minimal volume of the bounding box of the vertices. The connection to the problem of no three collinear points follows from the observation that a set V ⊂ Z 3 of n points induces a drawing of the complete graph K n if and only if no three points of V are collinear; for more details, see Pór and Wood [55]. Also, their open Problem 3 on vol (n, d, 1) is a question on dense d-dimensional point configurations without three points on a line, and their comment that this problem is trivial for d ≥ log 2 n follows from the trivial cap consisting of 2 d points with coordinates 0 and 1 only.…”

- Three-Dimensional Drawings

“…)[PW04]. Generalizing this construction, Pór and Wood[PW04] proved that if edge crossings are allowed, every c-colorable graph has a 3D drawing with O(n √ c) volume. That bound is optimal for the c-partite Turán graph.…”

Upward Three-Dimensional Grid Drawings of Graphs

“…3D drawings have been generalised in a number of ways. Multi-dimensional grid drawings have been studied [40,46], as have 3D polyline grid drawings, where edges are allowed to bend at gridpoints [6,17,18,37]. The focus of this paper is upward 3D drawings of directed graphs, which have previously been studied by Poranen [41] and Di Giacomo et al [10].…”