Weak saturation of tensor product of cliques

Given two hypergraphs $G$ and $H$, the weak saturation number $\operatorname{\mathrm{wsat}}(G,H)$ is the minimum number of edges in a spanning subhypergraph $F$ of $G$ such that the missing edges of $F$ can be added one at a time so that each added edge creates a copy of $H$. In this work, we determine weak saturation numbers for the case when $G$ and $H$ are tensor product of cliques, generalizing a result of Moshkovitz and Shapira (Journal of Combinatorial Theory, Series B, 2015), who found the exact values of $\operatorname{\mathrm{wsat}}(K^d_{n_1,\ldots,n_d},\ K^d_{r_1,\ldots,r_d})$. The proof also yields results for colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(G,H)$ of colored hypergraphs $G$ and $H$, where the colorings of the copies of $H$ must be compatible with the coloring of $G$. We determine these numbers when $G$ and $H$ are unions of tensor product of cliques, generalizing a result of Bulavka, Tancer, and Tyomkyn (Combinatorica, 2023), who determined $\operatorname{\mathrm{c-wsat}}(K^q_{n_1,\ldots,n_d}, K^q_{r_1,\ldots,r_d})$. Moreover, our proof allows us to generalize a result of Balogh, Bollob\'{a}s, Morris, and Riordan (Journal of Combinatorial Theory, Series A, 2012) by determining colored weak saturation numbers $\operatorname{\mathrm{c-wsat}}(K^d_{n_1,\ldots,n_d},\{K^d_{r_1,\ldots,r_d}\}_{\mathbf{r}\in \mathcal{R}})$ for an arbitrary family $\mathcal{R}$. The quantity $\operatorname{\mathrm{c-wsat}}(G,\mathcal{H})$ extends colored weak saturation by allowing, at each step, the creation of a colored copy of any hypergraph in the fixed family of hypergraphs $\mathcal{H}$.