{"paper":{"title":"A Ramsey Property of Random Regular and $k$-out Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Deepak Bal, Michael Anastos","submitted_at":"2017-08-03T16:56:51Z","abstract_excerpt":"In this note we consider a Ramsey property of random $d$-regular graphs, $\\mathcal{G}(n,d)$. Let $r\\ge 2$ be fixed. Then w.h.p. the edges of $\\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the other hand, there exists a constant $\\gamma > 0$ such that w.h.p., every $r$-coloring of the edges of $\\mathcal{G}(n, 2r+1)$ must contain a monochromatic cycle of length at least $\\gamma n$. We prove an analogous result for random $k$-out graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01211","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"}