Saturation Numbers for Berge Cliques

Abstract

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 $\varphi :E(F)\to E(\mathcal{H})$ such that $e\subseteq \varphi (e)$ 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 $k,\ell \ge 3$. Furthermore, we provide some sufficient conditions to imply that ${\mathrm{sat}}_k(n,\text{Berge-}F)=O(n)$ for general graphs $F$.

16 pages, 1 figure