Hypergraph saturation for the bow tie

TL;DR

This paper studies the minimal size of certain hypergraphs with no singleton intersections that become invalid upon adding any missing edge, providing tight bounds for the case when k=4, advancing understanding of hypergraph saturation.

Contribution

It establishes the first tight linear bounds for the hypergraph saturation problem at k=4, a significant step in the study of such combinatorial structures.

Findings

Established tight linear bounds for k=4 hypergraph saturation.

Extended the understanding of saturation and semi-saturation in hypergraphs.

Provided new insights into elementary cases of hypergraph intersection problems.

Abstract

Erd\H{o}s and S\'os initiated the study of the maximum size of a $k$-uniform set system, for $k \geq 4$, 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.