A linear upper bound for Berge saturation numbers

For a graph $F$ with at least one edge, a $k$-uniform hypergraph is Berge-$F$-saturated if it contains no Berge-$F$, but adding any missing hyperedge creates a Berge-$F$. The saturation number ${\text{sat}}_k(n,\text{Berge-}F)$ is the minimum number of edges in such a hypergraph on $n$ vertices. English, Gordon, Graber, Methuku and Sullivan conjectured a linear upper bound for every fixed finite family of forbidden graphs. We prove the single-graph case: ${\text{sat}}_k(n,\text{Berge-}F)=O_{F,k}(n)$ for every fixed integer $k\ge 2$ and every fixed finite simple graph $F$ with at least one edge. The proof combines a sparse construction with degree and matching arguments.