{"paper":{"title":"The asymptotic $\\chi$-boundedness of hereditary families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bruce Reed, Yelena Yuditsky","submitted_at":"2025-06-01T16:19:14Z","abstract_excerpt":"A family ${\\cal F}$ of graphs is asymptotically $\\chi$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $\\chi(G) \\le f(\\omega(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $\\chi$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $\\chi(G)=\\omega(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $\\chi(G) = \\omega(G)$. We show that the $C_6$-free graphs are asymptoti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01070","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.01070/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"}