{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SIVRXZO24BZ3WUMFWAEM6MMP2U","short_pith_number":"pith:SIVRXZO2","canonical_record":{"source":{"id":"1909.00514","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-09-02T02:16:15Z","cross_cats_sorted":[],"title_canon_sha256":"93ea43d2ad705b9a53fcd3c40ccdc8ef70f20a439432147a667820a3c70ce2d0","abstract_canon_sha256":"bf1c294c5885ed5e11d1348af4a0190a111986de92773cbe974c41419c49d007"},"schema_version":"1.0"},"canonical_sha256":"922b1be5dae073bb5185b008cf318fd514666b27fbea5c8675e14bf04d7c5424","source":{"kind":"arxiv","id":"1909.00514","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.00514","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1909.00514v3","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.00514","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"SIVRXZO24BZ3","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_16","alias_value":"SIVRXZO24BZ3WUMF","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_8","alias_value":"SIVRXZO2","created_at":"2026-07-05T01:39:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SIVRXZO24BZ3WUMFWAEM6MMP2U","target":"record","payload":{"canonical_record":{"source":{"id":"1909.00514","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-09-02T02:16:15Z","cross_cats_sorted":[],"title_canon_sha256":"93ea43d2ad705b9a53fcd3c40ccdc8ef70f20a439432147a667820a3c70ce2d0","abstract_canon_sha256":"bf1c294c5885ed5e11d1348af4a0190a111986de92773cbe974c41419c49d007"},"schema_version":"1.0"},"canonical_sha256":"922b1be5dae073bb5185b008cf318fd514666b27fbea5c8675e14bf04d7c5424","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:39:24.689662Z","signature_b64":"uqoFLU8C5n71sR5pI3/P7B4fr3D5x9I5Vy0lDJuP89LcyFR3bRWaX2682mqaVOzZBB8snEfW/3gZaPkW9WLfAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"922b1be5dae073bb5185b008cf318fd514666b27fbea5c8675e14bf04d7c5424","last_reissued_at":"2026-07-05T01:39:24.689217Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:39:24.689217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1909.00514","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s1ze60W75jLoswTAlILg+g608DyZYkUhlgBNQq06hs4fvC3Shx9tm4U49YY2ZirqaVjEHl0ETStSTeKifePqCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T07:05:18.795524Z"},"content_sha256":"d02fc0d15bc61789e931981b3ee9228ce749ce9ee39401671d99f729f93bb759","schema_version":"1.0","event_id":"sha256:d02fc0d15bc61789e931981b3ee9228ce749ce9ee39401671d99f729f93bb759"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SIVRXZO24BZ3WUMFWAEM6MMP2U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Progress towards Nash-Williams' Conjecture on Triangle Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Luke Postle, Michelle Delcourt","submitted_at":"2019-09-02T02:16:15Z","abstract_excerpt":"Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.\n  We show that for any graph "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.00514","kind":"arxiv","version":3},"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/1909.00514/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pUSrgVpJwZtGYXZZa2hWBRqVxXhgLn545lPfIPsbaogJHtS9Azmfy2yIR2Mc1r+X1KNsfbMHQXwr89FNV6TeAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T07:05:18.796213Z"},"content_sha256":"79be2f87c5107f60053fcd8c11ecac719c80da8402ffa134ccd26a8012ed620c","schema_version":"1.0","event_id":"sha256:79be2f87c5107f60053fcd8c11ecac719c80da8402ffa134ccd26a8012ed620c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/bundle.json","state_url":"https://pith.science/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/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":"2026-08-15T07:05:18Z","links":{"resolver":"https://pith.science/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U","bundle":"https://pith.science/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/bundle.json","state":"https://pith.science/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SIVRXZO24BZ3WUMFWAEM6MMP2U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SIVRXZO24BZ3WUMFWAEM6MMP2U","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":"bf1c294c5885ed5e11d1348af4a0190a111986de92773cbe974c41419c49d007","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-09-02T02:16:15Z","title_canon_sha256":"93ea43d2ad705b9a53fcd3c40ccdc8ef70f20a439432147a667820a3c70ce2d0"},"schema_version":"1.0","source":{"id":"1909.00514","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.00514","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1909.00514v3","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.00514","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"SIVRXZO24BZ3","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_16","alias_value":"SIVRXZO24BZ3WUMF","created_at":"2026-07-05T01:39:24Z"},{"alias_kind":"pith_short_8","alias_value":"SIVRXZO2","created_at":"2026-07-05T01:39:24Z"}],"graph_snapshots":[{"event_id":"sha256:79be2f87c5107f60053fcd8c11ecac719c80da8402ffa134ccd26a8012ed620c","target":"graph","created_at":"2026-07-05T01:39:24Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1909.00514/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.\n  We show that for any graph ","authors_text":"Luke Postle, Michelle Delcourt","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-09-02T02:16:15Z","title":"Progress towards Nash-Williams' Conjecture on Triangle Decompositions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.00514","kind":"arxiv","version":3},"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:d02fc0d15bc61789e931981b3ee9228ce749ce9ee39401671d99f729f93bb759","target":"record","created_at":"2026-07-05T01:39:24Z","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":"bf1c294c5885ed5e11d1348af4a0190a111986de92773cbe974c41419c49d007","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-09-02T02:16:15Z","title_canon_sha256":"93ea43d2ad705b9a53fcd3c40ccdc8ef70f20a439432147a667820a3c70ce2d0"},"schema_version":"1.0","source":{"id":"1909.00514","kind":"arxiv","version":3}},"canonical_sha256":"922b1be5dae073bb5185b008cf318fd514666b27fbea5c8675e14bf04d7c5424","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"922b1be5dae073bb5185b008cf318fd514666b27fbea5c8675e14bf04d7c5424","first_computed_at":"2026-07-05T01:39:24.689217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:39:24.689217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uqoFLU8C5n71sR5pI3/P7B4fr3D5x9I5Vy0lDJuP89LcyFR3bRWaX2682mqaVOzZBB8snEfW/3gZaPkW9WLfAA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:39:24.689662Z","signed_message":"canonical_sha256_bytes"},"source_id":"1909.00514","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d02fc0d15bc61789e931981b3ee9228ce749ce9ee39401671d99f729f93bb759","sha256:79be2f87c5107f60053fcd8c11ecac719c80da8402ffa134ccd26a8012ed620c"],"state_sha256":"e3ef7db3f033bd731e35dda72ec7eb0914635bd1ede560aa29d02bfa420a6b23"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ETs6HkL+KEc2FmEpgj+QxU91R792OIfw+rKM0N01XzPr7EA6pEJCUK1JwP7lNaZtCzVzJxg+OGO0azpCSrNRAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T07:05:18.801396Z","bundle_sha256":"8b18c582faff1ef8e3b9b410206b73929b6925b5d11ed1bcef8792a0027e0251"}}