Saturation for the 3-uniform loose 3-cycle

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∉E(H). The saturation number of F, denoted sat_k(n,F), is the minimum number of edges in an F-saturated k-uniform hypergraph H on n vertices. Let C₃⁽³⁾ denote the 3-uniform loose cycle on 3 edges. In this work, we prove that ([Formula presented]+o(1))n≤sat₃(n,C₃⁽³⁾)≤[Formula presented]n+O(1). This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.