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 $\mathrm{sat}_k(n,\text{Berge-}F)$ is the least number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show $$\mathrm{sat}_k(n,\text{Berge-}K_\ell)\sim \frac{\ell-2}{k-1}n,$$ for all $k,\ell\geq 3$. Furthermore, we provide some sufficient conditions to imply that $\mathrm{sat}_k(n,\text{Berge-}F)=O(n)$ for general graphs $F$.Comment: 16 pages, 1 figur

Linearity of Saturation for Berge Hypergraphs

For a graph F, we say a hypergraph H is Berge-F if it can be obtained from F be replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of Berge-F. The k-uniform saturation number of Berge-F, satk(n, Berge-F) is the fewest number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that satk(n, Berge-F) = O(n) for all graphs F and uniformities 3 ≤ k ≤ 5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraph