The general position number of integer lattices

The general position number $\mathrm{g}\mathrm{p}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three pairwise distinct vertices from $S$ lie on a common geodesic. The $n$-dimensional grid graph $\text{pn}$ is the Cartesian product of $n$ copies of the two-way infinite path $P_{\mathrm{\infty }}$. It is proved that if $n\in \mathbb{N}$, then ${\rm gp}({P_\infty^n}) = 2^{2^{n-1}}$\mathrm{.}Theresultwasearlierknownonlyfor$n\in {1,2}$andpartiallyfor$n=3$.