{"paper":{"title":"The extremal number of cycles with all diagonals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Benny Sudakov, Domagoj Brada\\v{c}","submitted_at":"2023-08-30T17:32:57Z","abstract_excerpt":"In 1975, Erd\\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\\H{o}s observed that the upper bound $O(n^{5/3})$ holds since the complete bipartite graph $K_{3,3}$ can be viewed as a cycle of length six with all diagonals.\n  In this paper, we resolve this old problem. We prove that there exists a constant $C$ such that every $n$-vertex with $Cn^{3/2}$ edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best pos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.16163","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/2308.16163/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"}