  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 $d\geq 2$ such that no three points are collinear satisfies the lower
bound \begin{align} \gg_d n^{d-1}\sqrt[2d]{d}.\nonumber \end{align}This pretty
much extends the result of the no-three-in-line problem to all dimension $d\geq
3$.
