Saturation numbers in tripartite graphs

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F⊈G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$\mathrm{.}Thesaturationnumberof$F$in$H$,denoted$$\text{sat}$(H,F)$,istheminimumnumberofedgesinan$F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\mathrm{\ell }\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\mathrm{\ell },\mathrm{\ell },\mathrm{\ell }})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\mathrm{\ell },\mathrm{\ell },\mathrm{\ell }-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3},K_{\mathrm{\ell },\mathrm{\ell },\mathrm{\ell }-2})$ within an additive constant. We also include general constructions of $K_{\mathrm{\ell },m,p}$-saturated subgraphs of $K_{n_1,n_2,n_3}$ with few edges for $\mathrm{\ell }\ge m\ge p>0$.