{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YTJ47UHYXLOOGA6BDPRBPVX54R","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b39391a0e2d2f1a3b6c4af1a22a17dd39e6959cd45ca93953a1579fa76084d96","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-26T03:42:58Z","title_canon_sha256":"d00e436bf923a5094a0d3ee7062effd42b6e0a3497340387cfe2a0fc7b1ca870"},"schema_version":"1.0","source":{"id":"1010.5309","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.5309","created_at":"2026-05-18T04:38:39Z"},{"alias_kind":"arxiv_version","alias_value":"1010.5309v1","created_at":"2026-05-18T04:38:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.5309","created_at":"2026-05-18T04:38:39Z"},{"alias_kind":"pith_short_12","alias_value":"YTJ47UHYXLOO","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YTJ47UHYXLOOGA6B","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YTJ47UHY","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:5e27cca7615edbe8ad69988755fd69f4968c021922b50ff8ad5515687065f991","target":"graph","created_at":"2026-05-18T04:38:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The {\\em independence ratio} of a graph $G$ is defined by \\[ \\iota(G) := \\sup_{X \\subset V(G)} \\frac{|X|}{\\alpha(X)},\\] where $\\alpha(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a relaxation of the chromatic number $\\chi(G)$ in the sense that $\\chi(G) \\geq \\iota(G)$ for every graph $G$, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erd\\H{o}s on cycles in graphs with large chromatic number and a conjecture of Erd\\H{o}s and Hajnal on graphs with infinite chromatic nu","authors_text":"Benny Sudakov, Jacques Verstraete","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-26T03:42:58Z","title":"Cycles in Sparse Graphs II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5309","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c63259e547f4e65d0f31d56ebe95b48becaa43df5365111b89ec02520da7e444","target":"record","created_at":"2026-05-18T04:38:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b39391a0e2d2f1a3b6c4af1a22a17dd39e6959cd45ca93953a1579fa76084d96","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-26T03:42:58Z","title_canon_sha256":"d00e436bf923a5094a0d3ee7062effd42b6e0a3497340387cfe2a0fc7b1ca870"},"schema_version":"1.0","source":{"id":"1010.5309","kind":"arxiv","version":1}},"canonical_sha256":"c4d3cfd0f8badce303c11be217d6fde4753ac799e258a80f3ceb65bedc9bfb9f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4d3cfd0f8badce303c11be217d6fde4753ac799e258a80f3ceb65bedc9bfb9f","first_computed_at":"2026-05-18T04:38:39.042276Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:39.042276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WQ3ubvcy5/WMDQlkDPvWAxBb/VcZMWY8YwX9tdvcPXpnJ2rkm7Sh4ahJNLaAqYMHpbt+yq+vulSTL1HyJJazAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:39.043062Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.5309","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c63259e547f4e65d0f31d56ebe95b48becaa43df5365111b89ec02520da7e444","sha256:5e27cca7615edbe8ad69988755fd69f4968c021922b50ff8ad5515687065f991"],"state_sha256":"e80f5b5693838600fe36f0b731630387b6ff9252ef4e34895b393d9e2a56a0a0"}