{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/hypergraph-saturation-for-the-bow-tie","title":"Hypergraph saturation for the bow tie","arxiv_id":"2408.16758","date":"2024-08-29","proceeding":null,"authors":["Stijn Cambie","Nika Salia"],"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.","url_abs":"https://arxiv.org/abs/2408.16758v1","url_pdf":"https://arxiv.org/pdf/2408.16758v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"hypergraph-saturation-for-the-bow-tie","repo_url":"https://github.com/stijncambie/hypergraphsaturation_bowtie","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}