{"paper":{"title":"Refined Tur\\'an numbers and Ramsey numbers for the loose 3-uniform path of length three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Ruci\\'nski, Joanna Polcyn","submitted_at":"2015-10-20T20:53:04Z","abstract_excerpt":"Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\\{a,b,c\\}, \\{c,d,e\\},$ and $\\{e,f,g\\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\\le 7$. The proof of this result relies on a careful analysis of the Tur\\'an numbers for $P$. In this paper, we refine this analysis further and compute, for all $n$, the third and fourth order Tur\\'an numbers for $P$. With the help of the former, we confirm the formula $R(P;r)=r+6$ for $r\\in\\{8,9\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06057","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"}