{"paper":{"title":"Bounded monochromatic components for random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Nicolas Broutin, Ross J. Kang","submitted_at":"2014-07-14T08:05:46Z","abstract_excerpt":"We consider vertex partitions of the binomial random graph $G_{n,p}$. For $np\\to\\infty$, we observe the following phenomenon: in any partition into asymptotically fewer than $\\chi(G_{n,p})$ parts, i.e. $o(np/\\log np)$ parts, one part must induce a connected component of order at least roughly the average part size.\n  Stated another way, we consider the $t$-component chromatic number, the smallest number of colours needed in a colouring of the vertices for which no monochromatic component has more than $t$ vertices. As long as $np \\to \\infty$, there is a threshold for $t$ around $\\Theta(p^{-1}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3555","kind":"arxiv","version":2},"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"}