Saturation in three-uniform Berge hypergraphs: the five-star spectrum and the $K_4$ saturation number

For a graph \(G\), a hypergraph is Berge-\(G\)-saturated if it contains no Berge copy of \(G\), but adding any missing hyperedge creates one. We determine the complete saturation spectrum of three-uniform Berge five-stars for every order. For \(n\ge7\), every integer from \(\lceil4n/3\rceil-3\) to \