On the general no-three-in-line problem

TL;DR

This paper extends the no-three-in-line problem to higher dimensions, establishing a lower bound on the maximum number of points in an n^d grid with no three collinear points.

Contribution

It generalizes the no-three-in-line problem to all dimensions d ≥ 3, providing a new lower bound for the maximum number of such points.

Findings

Established a lower bound of  n^{d-1} oot{2d}d for the problem.

Extended the classical 2D no-three-in-line result to higher dimensions.

Demonstrated the bound applies for all dimensions d  3.

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 $d\geq 3$.