Hypergraph saturation for the bow tie

Erdos and S\'os initiated the study of the maximum size of a k-uniform set system, for k 4, with no singleton intersections 50 years ago. In this work, we investigate the dual problem: finding the minimum size of a k-uniform hypergraph with no singleton intersections, such that adding any missing hyperedge forces a singleton intersection. These problems, known as saturation and semi-saturation, are typically challenging. Our focus is on an elementary-to-state case in the line of work by Erdos, F\"uredi and Tuza. We establish tight linear bounds for k=4, marking one of the first non-obvious cases with such a bound.