Linearity of Saturation for Berge Hypergraphs

TL;DR

This paper investigates the saturation number of Berge-$F$ hypergraphs, establishing linear bounds for uniformities 3 to 5, and extends the conjecture to hypergraph copies, advancing understanding in hypergraph saturation theory.

Contribution

It proves that the Berge-$F$-saturation number is linear in the number of vertices for uniformities 3 to 5, partially confirming a conjecture and extending it to hypergraph copies.

Findings

$ ext{sat}_k(n, ext{Berge-}F) = O(n)$ for all graphs $F$ and $3 \\leq k \\leq 5$

Partial confirmation of a conjecture on Berge hypergraph saturation numbers

Extension of the conjecture to Berge copies of hypergraphs

Abstract

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-$F$, $\mathrm{sat}_k(n,\text{Berge-}F)$ is the fewest number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show that $\mathrm{sat}_k(n,\text{Berge-}F) = O(n)$ for all graphs $F$ and uniformities $3\leq k\leq 5$, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraphs.