{"paper":{"title":"Cycles in triangle-free graphs of large chromatic number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Benny Sudakov, Jacques Verstraete","submitted_at":"2014-04-17T14:42:44Z","abstract_excerpt":"More than twenty years ago Erd\\H{o}s conjectured~\\cite{E1} that a triangle-free graph $G$ of chromatic number $k \\geq k_0(\\varepsilon)$ contains cycles of at least $k^{2 - \\varepsilon}$ different lengths as $k \\rightarrow \\infty$. In this paper, we prove the stronger fact that every triangle-free graph $G$ of chromatic number $k \\geq k_0(\\varepsilon)$ contains cycles of $(\\frac{1}{64} - \\varepsilon)k^2 \\log k$ consecutive lengths, and a cycle of length at least $(\\tfrac{1}{4} - \\varepsilon)k^2 \\log k$. As there exist triangle-free graphs of chromatic number $k$ with at most roughly $4k^2 \\log "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4544","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":""},"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"}