On general position sets in Cartesian products

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 distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\mathrm{\square }{C}_s$ is deduced. It is proved that $\mathrm{g}\mathrm{p}(C_r\mathrm{\square }{C}_s)\in \{6,7\}$ whenever $r\ge s\ge 3$, $s\mathrm{\ne }4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\mathrm{\square }{P}_s$, where $r,s\ge 2$, is also determined.