On the general no-three-in-line problem

Theophilus Agama

On the general no-three-in-line problem

Combinatorics 2026-04-14 v10 Metric Geometry

Authors: Theophilus Agama

Abstract

In this paper, we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n~(d~times)=n^d$ for all $d\in \mathbb{N}$ with $d\geq 2$ so that no three points are collinear satisfies the lower bound \begin{align} \gg n^{d-1}\sqrt[2d]{d}.\nonumber \end{align} This extends the result of the no-three-in-line problem to all dimension $d\geq 3$ .

Keywords

incidence geometry of points and lines

Cite

@article{arxiv.2106.15621,
  title  = {On the general no-three-in-line problem},
  author = {Theophilus Agama},
  journal= {arXiv preprint arXiv:2106.15621},
  year   = {2026}
}

Comments

11 pages; the paper has been reformatted and introduction expanded; ideas remain unchanged; arXiv admin note: substantial text overlap with arXiv:2006.05269, arXiv:1912.08075, arXiv:2002.00502

On the general no-three-in-line problem

Abstract

Keywords

Cite

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