The Saturation Spectrum of Berge Stars

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≥ 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 {Berge-F} if there exists a bijection φ:E(F)→ E(G) such that e⊆ φ(e) for all e∈ E(F). We completely determine the saturation spectrum for $3$-uniform Berge-K1, for 1≤ ≤ 4, and for =5 when 5 n. We also determine all but a constant number of values in the spectrum for $3$-uniform Berge-K1, for all ≥ 5. We note that this is the first result determining the saturation spectrum for any non-trivial hypergraph.