{"paper":{"title":"The Erd\\H{o}s-Hajnal Conjecture for Long Holes and Anti-holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Marthe Bonamy, Nicolas Bousquet, St\\'ephan Thomass\\'e","submitted_at":"2014-08-08T20:01:51Z","abstract_excerpt":"Erd\\H{o}s and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1964","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"}