{"paper":{"title":"A short proof that $\\chi$ can be bounded $\\epsilon$ away from $\\Delta+1$ towards $\\omega$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andrew D. King, Bruce A. Reed","submitted_at":"2012-11-06T22:02:27Z","abstract_excerpt":"In 1998 the second author proved that there is an $\\epsilon>0$ such that every graph satisfies $\\chi \\leq \\lceil (1-\\epsilon)(\\Delta+1)+\\epsilon\\omega\\rceil$. The first author recently proved that any graph satisfying $\\omega > \\frac 23(\\Delta+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Hall's Theorem, Brooks' Theorem, the Lov\\'asz Local Lemma, and Talagrand's Inequality"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1410","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"}