Papers › Hypergraph saturation for the bow tie
Hypergraph saturation for the bow tie
Stijn Cambie, Nika Salia
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Erd\H{o}s 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 Erd\H{o}s, F\"uredi and Tuza. We establish tight linear bounds for k=4, marking one of the first non-obvious cases with such a bound.
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