{"paper":{"title":"Many pentagons in triple systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Jozsef Solymosi","submitted_at":"2025-01-27T08:40:53Z","abstract_excerpt":"We prove that every $n$ vertex linear triple system with $m$ edges has at least $m^6/n^7$\n  copies of a pentagon, provided $m>100 \\, n^{3/2}$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan.\n  More generally, for each $ \\ell \\ge 2$, we prove that there is a constant $c$ such that if an $n$-vertex graph is $\\varepsilon$-far from being triangle-free, with $\\varepsilon \\gg n^{-1/3\\ell}$, then it has at least $c \\, \\varepsilon^{3\\ell} n^{2\\ell+1}$ copies of $C_{2\\ell+1}$. This improves the previous best bound of $c \\, \\varepsilon^{4\\ell+2} n^{2\\ell+1}$ due to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.15861","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.15861/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}