Saturation for the $3$-uniform loose $3$-cycle

TL;DR

This paper establishes bounds on the minimum number of edges in a 3-uniform hypergraph that is saturated with respect to the loose 3-cycle, marking the first such result for a fixed short hypergraph cycle.

Contribution

It provides the first non-trivial bounds on the saturation number for the 3-uniform loose 3-cycle, advancing understanding in hypergraph saturation theory.

Findings

Lower bound: approximately (4/3)n edges

Upper bound: approximately (3/2)n edges

First non-trivial bounds for a fixed short hypergraph cycle

Abstract

Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is the minimum number of edges in a $F$-saturated $k$-uniform hypergraph $H$ on $n$ vertices. Let $C_3^{(3)}$ denote the $3$-uniform loose cycle on $3$ edges. In this work, we prove that \[ \left(\frac{4}3+o(1)\right)n\leq \mathrm{sat}_3(n,C_3^{(3)})\leq \frac{3}2n+O(1). \] This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.