{"paper":{"title":"The extremal function for cycles of length $\\ell$ mod $k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Jacques Verstraete","submitted_at":"2016-06-28T01:47:50Z","abstract_excerpt":"Burr and Erd\\H{o}s conjectured that for each $k,\\ell \\in \\mathbb Z^+$ such that $k \\mathbb Z + \\ell$ contains even integers, there exists $c_k(\\ell)$ such that any graph of average degree at least $c_k(\\ell)$ contains a cycle of length $\\ell$ mod $k$. This conjecture was proved by Bollob\\'{a}s, and many successive improvements of upper bounds on $c_k(\\ell)$ appear in the literature. In this short note, for $1 \\leq \\ell \\leq k$, we show that $c_k(\\ell)$ is proportional to the largest average degree of a $C_{\\ell}$-free graph on $k$ vertices, which determines $c_k(\\ell)$ up to an absolute consta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08532","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"}