{"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/the-turan-density-of-short-tight-cycles","title":"The Turán density of short tight cycles","arxiv_id":"2506.03223","date":"2025-06-03","proceeding":null,"authors":["Levente Bodnár","Jared León","Xizhi Liu","Oleg Pikhurko"],"abstract":"The $3$-uniform tight $\\ell$-cycle $C_\\ell^{3}$ is the $3$-graph on $\\{1,\\dots,\\ell\\}$ consisting of all $\\ell$ consecutive triples in the cyclic order. Let $\\mathcal{C}$ be either the pair $\\{C_{4}^{3}, C_{5}^{3}\\}$ or the single tight $\\ell$-cycle $C_{\\ell}^{3}$ for some $\\ell\\ge 7$ not divisible by $3$. We show that the Tur\\'an density of $\\mathcal{C}$, that is, the asymptotically maximal edge density of a large $\\mathcal{C}$-free $3$-graph, is equal to $2\\sqrt{3} - 3$. We also establish the corresponding Erd\\H{o}s-Simonovits-type stability result, informally stating that all almost maximum $\\mathcal{C}$-free graphs are close in the edit distance to a 2-part recursive construction. This extends the earlier analogous results of Kam\\v{c}ev-Letzter-Pokrovskiy [\"The Tur\\'an density of tight cycles in three-uniform hypergraphs\", Int. Math. Res. Not. 6 (2024), 4804-4841] that apply for sufficiently large $\\ell$ only. Additionally, we prove a finer structural result that allows us to determine the maximum number of edges in a $\\{C_{4}^{3}, C_{5}^{3}\\}$-free $3$-graph with a given number of vertices up to an additive $O(1)$ error term.","url_abs":"https://arxiv.org/abs/2506.03223v1","url_pdf":"https://arxiv.org/pdf/2506.03223v1.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":"the-turan-density-of-short-tight-cycles","repo_url":"https://github.com/bodnalev/supplementary_files","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}