The Saturation Spectrum of Berge Stars

TL;DR

This work studies the saturation spectrum for Berge-$F$ in 3-uniform hypergraphs, focusing on Berge-stars $\text{Berge-}K_{1,\ell}$. It develops a Berge-degree framework via vertex links, introduces aggressive saturation notions, and employs a configuration-model-based random construction to realize prescribed degree patterns, enabling modular saturated graphs. The authors prove a near-complete spectrum for $\ell\ge5$, with a full determination for $\ell=5$ when $5\mid n$, and completely determine the spectrum for $\ell\le4$, marking the first full spectrum results for non-trivial hypergraphs. The results are achieved through lantern/sun gadget constructions, disjoint unions with dense cores, and carefully tuned gadget families that interpolate edge counts, offering a toolkit for future saturation-spectrum studies in hypergraphs and higher uniformities.

Abstract

The forbidden subgraph problem is among the oldest in extremal combinatorics -- how many edges can an $n$-vertex $F$-free graph have? The answer to this question is the well-studied extremal number of $F$. Observing that every extremal example must be maximally $F$-free, a natural minimization problem is also studied -- how few edges can an $n$-vertex maximal $F$-free graph have? This leads to the saturation number of $F$. Both of these problems are notoriously difficult to extend to $k$-uniform hypergraphs for any $k\ge 3$. Barefoot et al., in the case of forbidding triangles in graphs, asked a beautiful question -- which numbers of edges, between the saturation number and the extremal number, are actually realized by an $n$-vertex maximal $F$-free graph? Hence named the saturation spectrum of $F$, this has since been determined precisely for several classes of graphs through a large number of papers over the past two decades. In this paper, we extend the notion of the saturation spectrum to the hypergraph context. Given a graph $F$ and a hypergraph $G$ embedded on the same vertex set, we say $G$ is a {\bf{Berge-$F$}} if there exists a bijection $φ:E(F)\to E(G)$ such that $e\subseteq φ(e)$ for all $e\in E(F)$. We completely determine the saturation spectrum for $3$-uniform Berge-$K_{1,\ell}$ for $1\leq \ell\leq 4$, and for $\ell=5$ when $5\mid n$. We also determine all but a constant number of values in the spectrum for $3$-uniform Berge-$K_{1,\ell}$ for all $\ell\geq 5$. We note that this is the first result determining the saturation spectrum for any non-trivial hypergraph.

Figures (3)

• Figure 1: Diagrams of some of the constructions required for the proofs of Theorems \ref{['theorem lower range']}, \ref{['theorem upper range']} and \ref{['theorem upper range for L=5']}. A dashed rectangle represents that the $2$-graph inside the rectangle is part of the link of any vertex connected to the rectangle via dashed lines. All smooth curves represent $3$-edges in the graph.
• Figure 2: All allowable links of vertices $v$ with $|N(v)|\geq 5$ in a Berge-$K_{1,5}$-saturated graph.
• Figure 3: Diagrams of some of the constructions required for the proof of Theorem \ref{['theorem upper range for L=5']}. A dashed rectangle represents that the $2$-graph inside the rectangle is part of the link of any vertex connected to the rectangle via dashed lines. All smooth curves represent $3$-edges in the graph.

Theorems & Definitions (61)

• Theorem 1.1: Theorem 12 in GMP
• Theorem 1.2: Theorem 4 in AE
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• proof
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3