On saturation of Berge hypergraphs

A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain any Berge copies of $F$. We address the saturation problem concerning Berge-$F$-free hypergraphs, i.e., what is the minimum number $sat_r(n,F)$ of hyperedges in an $r$-uniform Berge-$F$-free hypergraph $H$ with the property that adding any new hyperedge to $H$ creates a Berge copy of $F$. We prove that $sat_r(n,F)$ grows linearly in $n$ if $F$ is either complete multipartite or it possesses the following property: if $d_1\le d_2\le \dots \le d_{|V(F)|}$ is the degree sequence of $F$, then $F$ contains two adjacent vertices $u,v$ with $d_F(u)=d_1$, $d_F(v)=d_2$. In particular, the Berge-saturation number of regular graphs grows linearly in $n$.