{"paper":{"title":"Planar Tur\\'{a}n Numbers of Cycles: A Counterexample","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhinav Shantanam, Bernard Lidick\\'y, Daniel W. Cranston, Xiaonan Liu","submitted_at":"2021-10-05T13:48:47Z","abstract_excerpt":"The planar Turan number $\\textrm{ex}_{\\mathcal{P}}(C_{\\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\\ell$-cycle. For $\\ell\\in \\{3,4,5,6\\}$, upper bounds on $\\textrm{ex}_{\\mathcal{P}}(C_{\\ell},n)$ are known that hold with equality infinitely often. Ghosh, Gy\\\"{o}ri, Martin, Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on $\\textrm{ex}_{\\mathcal{P}}(C_{\\ell},n)$ for every $\\ell\\ge 7$ and $n$ sufficiently large. We disprove this conjecture for every $\\ell\\ge 11$. We also propose two revised versions of the conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.02043","kind":"arxiv","version":1},"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/2110.02043/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"}