{"paper":{"title":"Chromatic thresholds in sparse random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julia B\\\"ottcher, Peter Allen, Robert Morris, Simon Griffiths, Yoshiharu Kohayakawa","submitted_at":"2015-08-16T22:10:10Z","abstract_excerpt":"The chromatic threshold $\\delta_\\chi(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \\subset G(n,p)$ with minimum degree $\\delta(G) \\ge dpn$ has bounded chromatic number. The study of $\\delta_\\chi(H) :=\\delta_\\chi(H,1)$ was initiated in 1973 by Erd\\H{o}s and Simonovits. Recently $\\delta_\\chi(H)$ was determined for all graphs $H$. It is known that $\\delta_\\chi(H,p) =\\delta_\\chi(H)$ for all fixed $p \\in (0,1)$, but that typically $\\delta_\\chi(H,p) \\ne \\delta_\\chi(H)$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03875","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"}