Saturation Numbers for Berge Cliques

TL;DR

This paper investigates the minimum size of hypergraphs that are saturated with respect to Berge copies of a graph, establishing asymptotic formulas for complete graphs and conditions for linear bounds.

Contribution

It introduces the concept of Berge-$F$-saturation in hypergraphs and derives asymptotic saturation numbers for Berge-$K_ ext{ell}$, along with conditions for linear bounds in general graphs.

Findings

Asymptotic saturation number for Berge-$K_ ext{ell}$ is approximately rac{ ext{ell}-2}{k-1}n.

Provides sufficient conditions for the saturation number to be linear in n for general graphs.

Establishes a framework connecting Berge hypergraph saturation to classical graph saturation concepts.

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 $\phi:E(F)\to E(\mathcal{H})$ 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 $F$.