The Saturation Spectrum of Berge Stars

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 $\phi:E(F)\to E(G)$ such that $e\subseteq \phi(e)$ 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.