{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:APUNC4SNG6SJLMLNEMHWE37SHB","short_pith_number":"pith:APUNC4SN","canonical_record":{"source":{"id":"1011.2500","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-10T21:02:26Z","cross_cats_sorted":[],"title_canon_sha256":"c32fa3efefc47e7bccefe66653e354c58b1a8d84a754e53d82ad695776c7fe68","abstract_canon_sha256":"f10035d5636f4d09f07b966e2769cfa2e1e589adcc2a198e19ebeea090cb6257"},"schema_version":"1.0"},"canonical_sha256":"03e8d1724d37a495b16d230f626ff238436ab62e53045258243d8fa6816e47d1","source":{"kind":"arxiv","id":"1011.2500","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.2500","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"arxiv_version","alias_value":"1011.2500v2","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2500","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"pith_short_12","alias_value":"APUNC4SNG6SJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"APUNC4SNG6SJLMLN","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"APUNC4SN","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:APUNC4SNG6SJLMLNEMHWE37SHB","target":"record","payload":{"canonical_record":{"source":{"id":"1011.2500","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-10T21:02:26Z","cross_cats_sorted":[],"title_canon_sha256":"c32fa3efefc47e7bccefe66653e354c58b1a8d84a754e53d82ad695776c7fe68","abstract_canon_sha256":"f10035d5636f4d09f07b966e2769cfa2e1e589adcc2a198e19ebeea090cb6257"},"schema_version":"1.0"},"canonical_sha256":"03e8d1724d37a495b16d230f626ff238436ab62e53045258243d8fa6816e47d1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:11.084028Z","signature_b64":"fwXHBFPqT9gMz+0ni1WSDEQvfcaZyPab4jPVw7U48kcdYUhDgYJfFqxHHo6/R2Tmal3qD563Wa+p2IWUbLSNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03e8d1724d37a495b16d230f626ff238436ab62e53045258243d8fa6816e47d1","last_reissued_at":"2026-05-18T03:50:11.083442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:11.083442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.2500","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+ata+4l9HN9eJwOT6ZzzM9pPe/0meftsY93D5FbKeDoYxfcH36vXEjOYsRGdnGuVgiiUnSGfTWv8LvOvbScFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:31:53.503960Z"},"content_sha256":"3d674fa667661e8dd46f1367991b328190c42b525c7425aa7cf8b6ee7af6d4f4","schema_version":"1.0","event_id":"sha256:3d674fa667661e8dd46f1367991b328190c42b525c7425aa7cf8b6ee7af6d4f4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:APUNC4SNG6SJLMLNEMHWE37SHB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Fractional Chromatic Number of Triangle-free Graphs with $\\Delta\\leq 3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Xing Peng","submitted_at":"2010-11-10T21:02:26Z","abstract_excerpt":"Let $G$ be any triangle-free graph with maximum degree $\\Delta\\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of $G$, namely $\\chi_f(G)\\leq 14/5. Recently, Hatami and Zhu proved $\\chi_f(G) \\leq 3 -{3/64}$. In this paper, we prove $\\chi_f(G) \\leq 3- 3/43$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2500","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Fx82b2T4bFxYgty4v60Oo0w/4EEWGj7/fCvsc5oLAlJJoLm7zY1PgVTIuDxmIx3Y76hdo2RLqhDk4EwRy2NAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:31:53.504527Z"},"content_sha256":"6c0b9b6ba2664aab6c1060bf820c4b8801f6812667c52cda22ae680fa5d6cad3","schema_version":"1.0","event_id":"sha256:6c0b9b6ba2664aab6c1060bf820c4b8801f6812667c52cda22ae680fa5d6cad3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/APUNC4SNG6SJLMLNEMHWE37SHB/bundle.json","state_url":"https://pith.science/pith/APUNC4SNG6SJLMLNEMHWE37SHB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/APUNC4SNG6SJLMLNEMHWE37SHB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T23:31:53Z","links":{"resolver":"https://pith.science/pith/APUNC4SNG6SJLMLNEMHWE37SHB","bundle":"https://pith.science/pith/APUNC4SNG6SJLMLNEMHWE37SHB/bundle.json","state":"https://pith.science/pith/APUNC4SNG6SJLMLNEMHWE37SHB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/APUNC4SNG6SJLMLNEMHWE37SHB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:APUNC4SNG6SJLMLNEMHWE37SHB","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":"f10035d5636f4d09f07b966e2769cfa2e1e589adcc2a198e19ebeea090cb6257","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-10T21:02:26Z","title_canon_sha256":"c32fa3efefc47e7bccefe66653e354c58b1a8d84a754e53d82ad695776c7fe68"},"schema_version":"1.0","source":{"id":"1011.2500","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.2500","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"arxiv_version","alias_value":"1011.2500v2","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2500","created_at":"2026-05-18T03:50:11Z"},{"alias_kind":"pith_short_12","alias_value":"APUNC4SNG6SJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"APUNC4SNG6SJLMLN","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"APUNC4SN","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:6c0b9b6ba2664aab6c1060bf820c4b8801f6812667c52cda22ae680fa5d6cad3","target":"graph","created_at":"2026-05-18T03:50:11Z","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":"Let $G$ be any triangle-free graph with maximum degree $\\Delta\\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of $G$, namely $\\chi_f(G)\\leq 14/5. Recently, Hatami and Zhu proved $\\chi_f(G) \\leq 3 -{3/64}$. In this paper, we prove $\\chi_f(G) \\leq 3- 3/43$.","authors_text":"Linyuan Lu, Xing Peng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-10T21:02:26Z","title":"The Fractional Chromatic Number of Triangle-free Graphs with $\\Delta\\leq 3$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2500","kind":"arxiv","version":2},"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":"sha256:3d674fa667661e8dd46f1367991b328190c42b525c7425aa7cf8b6ee7af6d4f4","target":"record","created_at":"2026-05-18T03:50:11Z","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":"f10035d5636f4d09f07b966e2769cfa2e1e589adcc2a198e19ebeea090cb6257","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-10T21:02:26Z","title_canon_sha256":"c32fa3efefc47e7bccefe66653e354c58b1a8d84a754e53d82ad695776c7fe68"},"schema_version":"1.0","source":{"id":"1011.2500","kind":"arxiv","version":2}},"canonical_sha256":"03e8d1724d37a495b16d230f626ff238436ab62e53045258243d8fa6816e47d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03e8d1724d37a495b16d230f626ff238436ab62e53045258243d8fa6816e47d1","first_computed_at":"2026-05-18T03:50:11.083442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:11.083442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fwXHBFPqT9gMz+0ni1WSDEQvfcaZyPab4jPVw7U48kcdYUhDgYJfFqxHHo6/R2Tmal3qD563Wa+p2IWUbLSNBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:11.084028Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.2500","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d674fa667661e8dd46f1367991b328190c42b525c7425aa7cf8b6ee7af6d4f4","sha256:6c0b9b6ba2664aab6c1060bf820c4b8801f6812667c52cda22ae680fa5d6cad3"],"state_sha256":"7d568be2d9208e756750f63a39caaf2f55402c86047969c1e70c6ced9d4cc56a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3bDGWZLPlLPlhr6kIb3oU869KXvp0WR1k+xW39qHHunqXpKZYwcZqfWmI9tEMQOtILaY8bne07CLsJAuuuxpBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T23:31:53.507650Z","bundle_sha256":"0592e0ed05010981b1ad36274e3c6454e9b9c686743b0850b6e03a80b80ac1ca"}}