Saturation Numbers for Berge Cliques

Let $F$ be a graph and $\mathcal{H}$ be a hypergraph, both embedded on the same vertex set. We say $\mathcal{H}$ is a Berge-$F$ if there exists a bijection $\phi :E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi (e)$ for all $e\in E(F)$. We say $\mathcal{H}$ is Berge-$F$-saturated if $\mathcal{H}$ does not contain any Berge-$F$, but adding any missing edge to $\mathcal{H}$ creates a copy of a Berge-$F$. The saturation number ${\mathrm{s}\mathrm{a}\mathrm{t}}_k(n,\text{Berge-}F)$ is the least number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show $${\mathrm{s}\mathrm{a}\mathrm{t}}_k(n,\text{Berge-}{K}_{\mathrm{\ell }})\sim \frac{\mathrm{\ell }-2}{k-1}n,$$ for all $k,\mathrm{\ell }\ge 3$. Furthermore, we provide some sufficient conditions to imply that ${\mathrm{s}\mathrm{a}\mathrm{t}}_k(n,\text{Berge-}F)=O(n)$ for general graphs $F$.