Hypergraph saturation for the bow tie
Abstract
Erdős and Sós 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 Erdős, Füredi and Tuza. We establish tight linear bounds for $k=4$, marking one of the first non-obvious cases with such a bound.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.16758
- arXiv:
- arXiv:2408.16758
- Bibcode:
- 2024arXiv240816758C
- Keywords:
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- Mathematics - Combinatorics;
- 05C35;
- 05C65;
- 05C70
- E-Print:
- 11 pages, 2 figures