{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:JYIDM7357RB262L6T3QO5UFOCN","short_pith_number":"pith:JYIDM735","canonical_record":{"source":{"id":"1308.1872","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-08T15:12:48Z","cross_cats_sorted":[],"title_canon_sha256":"943faaec654a2f744bea65316238ea17c4b061458e5cdac0e4f76c94ece0a361","abstract_canon_sha256":"43237e6ad9be88d5d12a9bd9239216300e94e2ca9414390438264206d2e7f1a2"},"schema_version":"1.0"},"canonical_sha256":"4e10367f7dfc43af697e9ee0eed0ae1357dfdc125c9ecc803a45adc236c9ec1f","source":{"kind":"arxiv","id":"1308.1872","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1872","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1872v2","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1872","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"pith_short_12","alias_value":"JYIDM7357RB2","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JYIDM7357RB262L6","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JYIDM735","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:JYIDM7357RB262L6T3QO5UFOCN","target":"record","payload":{"canonical_record":{"source":{"id":"1308.1872","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-08T15:12:48Z","cross_cats_sorted":[],"title_canon_sha256":"943faaec654a2f744bea65316238ea17c4b061458e5cdac0e4f76c94ece0a361","abstract_canon_sha256":"43237e6ad9be88d5d12a9bd9239216300e94e2ca9414390438264206d2e7f1a2"},"schema_version":"1.0"},"canonical_sha256":"4e10367f7dfc43af697e9ee0eed0ae1357dfdc125c9ecc803a45adc236c9ec1f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:46.099160Z","signature_b64":"GVqIzEqLDXYeBHaoe3Jjoynwx67WHWEJWGFI9dwj9b7hbEHaDCByj2hacBOk/tDJyp6hhnsiFZ50bcxQy2/RCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e10367f7dfc43af697e9ee0eed0ae1357dfdc125c9ecc803a45adc236c9ec1f","last_reissued_at":"2026-05-18T01:04:46.098492Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:46.098492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.1872","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":"2026-05-18T01:04:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VG/cbjbpuZeVfO5CDBKXw/aekdfFfM4eZGRNfQR/MEeWoUOkiIiF/UAVMNXS9j0dLhvIEO8VlXmSPj3RFASaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:28:40.465771Z"},"content_sha256":"c6892de3d32f5041f8abb620b0eb64226cc022a216a7b746dd140d8c2746e0de","schema_version":"1.0","event_id":"sha256:c6892de3d32f5041f8abb620b0eb64226cc022a216a7b746dd140d8c2746e0de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:JYIDM7357RB262L6T3QO5UFOCN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the chromatic number of random Cayley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Green","submitted_at":"2013-08-08T15:12:48Z","abstract_excerpt":"Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime.\n  Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1872","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":"2026-05-18T01:04:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"82RL5K9zDCUwBYNQzcN2LJVhMtHck6yCWy+7NJxJD3kkuC82oFoToTkKygRKZ+lFAIFr+pncnOvkSmPJmFqCDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:28:40.466125Z"},"content_sha256":"d620637d72a6f56003d7cd233b3d04e81cf8d8af7129f6e4f7fa081b397ecc60","schema_version":"1.0","event_id":"sha256:d620637d72a6f56003d7cd233b3d04e81cf8d8af7129f6e4f7fa081b397ecc60"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JYIDM7357RB262L6T3QO5UFOCN/bundle.json","state_url":"https://pith.science/pith/JYIDM7357RB262L6T3QO5UFOCN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JYIDM7357RB262L6T3QO5UFOCN/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":"2026-06-01T03:28:40Z","links":{"resolver":"https://pith.science/pith/JYIDM7357RB262L6T3QO5UFOCN","bundle":"https://pith.science/pith/JYIDM7357RB262L6T3QO5UFOCN/bundle.json","state":"https://pith.science/pith/JYIDM7357RB262L6T3QO5UFOCN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JYIDM7357RB262L6T3QO5UFOCN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:JYIDM7357RB262L6T3QO5UFOCN","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":"43237e6ad9be88d5d12a9bd9239216300e94e2ca9414390438264206d2e7f1a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-08T15:12:48Z","title_canon_sha256":"943faaec654a2f744bea65316238ea17c4b061458e5cdac0e4f76c94ece0a361"},"schema_version":"1.0","source":{"id":"1308.1872","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1872","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1872v2","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1872","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"pith_short_12","alias_value":"JYIDM7357RB2","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JYIDM7357RB262L6","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JYIDM735","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:d620637d72a6f56003d7cd233b3d04e81cf8d8af7129f6e4f7fa081b397ecc60","target":"graph","created_at":"2026-05-18T01:04:46Z","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 an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime.\n  Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013.","authors_text":"Ben Green","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-08T15:12:48Z","title":"On the chromatic number of random Cayley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1872","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:c6892de3d32f5041f8abb620b0eb64226cc022a216a7b746dd140d8c2746e0de","target":"record","created_at":"2026-05-18T01:04:46Z","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":"43237e6ad9be88d5d12a9bd9239216300e94e2ca9414390438264206d2e7f1a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-08-08T15:12:48Z","title_canon_sha256":"943faaec654a2f744bea65316238ea17c4b061458e5cdac0e4f76c94ece0a361"},"schema_version":"1.0","source":{"id":"1308.1872","kind":"arxiv","version":2}},"canonical_sha256":"4e10367f7dfc43af697e9ee0eed0ae1357dfdc125c9ecc803a45adc236c9ec1f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e10367f7dfc43af697e9ee0eed0ae1357dfdc125c9ecc803a45adc236c9ec1f","first_computed_at":"2026-05-18T01:04:46.098492Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:46.098492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GVqIzEqLDXYeBHaoe3Jjoynwx67WHWEJWGFI9dwj9b7hbEHaDCByj2hacBOk/tDJyp6hhnsiFZ50bcxQy2/RCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:46.099160Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.1872","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6892de3d32f5041f8abb620b0eb64226cc022a216a7b746dd140d8c2746e0de","sha256:d620637d72a6f56003d7cd233b3d04e81cf8d8af7129f6e4f7fa081b397ecc60"],"state_sha256":"b9c3a427fb43e857e532f5e8b9c520b5f311c3795b008c038328a0e2d6963d56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SKq62jO/vl0X1k8cFkhhtWSHsV1t6n5oo5TZPn2/QGycouyq9LHThRHEJ4+r2ym+DRWmfdS6UA4CGnH/E6fFAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:28:40.468351Z","bundle_sha256":"fdb40cd99565c6f086f9839479f017386625e30588876424fdfc0f4a97b7b3d6"}}