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 not in E(H). The saturation number of F, denoted sat_k(n,F), is the minimum number of edges in a F-saturated k-uniform hypergraph on n vertices. In this talk, we will give a brief history of the saturation problem for cycles in graphs and hypergraphs, and then we will sketch a proof that
4n/3+o(n) \leq sat_3(n,C_3^3) \leq 3n/2+O(1),
where C_3^3 is the 3-uniform loose cycle on 3 edges. This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.
This project was joint work with Alexandr Kostochka and Dara Zirlin.