Total mutual-visibility in Hamming graphs

If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P)\cap X\subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number ${\mu }_{\mathrm{t}}(G)$ of $G$. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values ${\mu }_{\mathrm{t}}(K_{n_1}\mathrm{\square }{K}_{n_2}\mathrm{\square }{K}_{n_3})$ are determined. It is proved that ${\mu }_{\mathrm{t}}(K_{n_1}\mathrm{\square }\cdots \mathrm{\square }{K}_{n_r})=O(N^{r-2})$, where $N=n_1+\cdots +n_r$, and that ${\mu }_{\mathrm{t}}(K_s^{\mathrm{\square },r})=\mathrm{\Theta }(s^{r-2})$ for every $r\ge 3$, where $K_s^{\mathrm{\square },r}$ denotes the Cartesian product of $r$ copies of $K_s$. The main theorems are also reformulated as Turán-type results on hypergraphs.