{"paper":{"title":"Cycle-complete Ramsey numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eoin Long, Jozef Skokan, Peter Keevash","submitted_at":"2018-07-17T12:17:40Z","abstract_excerpt":"The Ramsey number $r(C_{\\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\\ell$ or a blue clique of order $n$. In 1978, Erd\\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\\ell},K_n) = (\\ell-1)(n-1)+1$ for $\\ell \\geq n\\geq 3$ provided $(\\ell,n) \\neq (3,3)$.\n  We prove that, for some absolute constant $C\\ge 1$, we have $r(C_{\\ell},K_n) = (\\ell-1)(n-1)+1$ provided $\\ell \\geq C\\frac {\\log n}{\\log \\log n}$. Up to the value of $C$ this is tight since we also show that, for any $\\varepsilon >0$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06376","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"}