{"paper":{"title":"Induced subgraph density. II. Sparse and dense sets in cographs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Scott, Jacob Fox, Paul Seymour, Tung Nguyen","submitted_at":"2023-07-03T07:33:36Z","abstract_excerpt":"A well-known theorem of R\\\"odl says that for every graph $H$, and every $\\epsilon>0$, there exists $\\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\\subseteq V(G)$ with $|X|\\ge \\delta|G|$ such that one of $G[X],\\overline{G}[X]$ has edge-density at most $\\epsilon$. But how does $\\delta$ depend on $\\epsilon$? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all $H$ there exists $c>0$ such that for all $\\epsilon$ with $0<\\epsilon\\le 1/2$, R\\\"odl's theorem holds with $\\delta=\\epsilon^c$. This conjecture implies the Erd\\H{o}s-Ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.00801","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.00801/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}