{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:IHETMTKKQU2X7CAJ7J5RZ2MGJO","short_pith_number":"pith:IHETMTKK","canonical_record":{"source":{"id":"2312.09202","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-14T18:32:20Z","cross_cats_sorted":[],"title_canon_sha256":"ecf08ca192cbe43b9ac2289dd42aad78cfc2f10533b99b94d728ffcd58424e07","abstract_canon_sha256":"f12af5e2252607aad1c4b8d0bcf2842b512fe3962979e1d83089c219002291ae"},"schema_version":"1.0"},"canonical_sha256":"41c9364d4a85357f8809fa7b1ce9864baf2c093d6bab0c606846038aec8f70cf","source":{"kind":"arxiv","id":"2312.09202","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2312.09202","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"arxiv_version","alias_value":"2312.09202v2","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.09202","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_12","alias_value":"IHETMTKKQU2X","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_16","alias_value":"IHETMTKKQU2X7CAJ","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_8","alias_value":"IHETMTKK","created_at":"2026-07-05T11:52:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:IHETMTKKQU2X7CAJ7J5RZ2MGJO","target":"record","payload":{"canonical_record":{"source":{"id":"2312.09202","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-14T18:32:20Z","cross_cats_sorted":[],"title_canon_sha256":"ecf08ca192cbe43b9ac2289dd42aad78cfc2f10533b99b94d728ffcd58424e07","abstract_canon_sha256":"f12af5e2252607aad1c4b8d0bcf2842b512fe3962979e1d83089c219002291ae"},"schema_version":"1.0"},"canonical_sha256":"41c9364d4a85357f8809fa7b1ce9864baf2c093d6bab0c606846038aec8f70cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:52:51.712086Z","signature_b64":"XebpEPcHMNdsmLGiytJOJL3JHNRiKjTEW3BVjPaedO3VUcF7q/G2SoHIunmer2MxYuF3gZCQNBXP4p9fH7V/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41c9364d4a85357f8809fa7b1ce9864baf2c093d6bab0c606846038aec8f70cf","last_reissued_at":"2026-07-05T11:52:51.711603Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:52:51.711603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2312.09202","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":"2026-07-05T11:52:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yYRYGuKqKlKIs93ZApK2Tq1Db5MM6sJV0J6h7KNZAi1TRRw+26vFQdsWVvzFmHXEb2ITcYknAqDCqSOObcHZCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T18:18:25.517770Z"},"content_sha256":"a7b0477518b0d744d95f9e194c72149caeb16eafdb848253b925c0db11c724a4","schema_version":"1.0","event_id":"sha256:a7b0477518b0d744d95f9e194c72149caeb16eafdb848253b925c0db11c724a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:IHETMTKKQU2X7CAJ7J5RZ2MGJO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the evolution of structure in triangle-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aditya Potukuchi, Matthew Jenssen, Will Perkins","submitted_at":"2023-12-14T18:32:20Z","abstract_excerpt":"We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.\n  Erd\\H{o}s, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite. Osthus, Pr\\\"omel, and Taraz later showed that for $m \\ge (1+\\epsilon)\\frac{\\sqrt{3}}{4}n^{3/2}\\sqrt{\\log n}$, almost every triangle-free graph on $n$ vertices and $m$ edges is bipartite. Here we give a precise characterization of the distribution of edges within ea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.09202","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.09202/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":"2026-07-05T11:52:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7UzMmPIKNiZ2M+jYZamjzjMsaINXtE7/sucHhHkTFUBZXuSx+NQx9bTs5gIpD26Di0g4/2YL53qaoNb7tungDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T18:18:25.518592Z"},"content_sha256":"fccdb9bc84bd421bb7808cbd9f5138eca1a36cf0f0e3790ca37a91429805a22f","schema_version":"1.0","event_id":"sha256:fccdb9bc84bd421bb7808cbd9f5138eca1a36cf0f0e3790ca37a91429805a22f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/bundle.json","state_url":"https://pith.science/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/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":"2026-08-17T18:18:25Z","links":{"resolver":"https://pith.science/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO","bundle":"https://pith.science/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/bundle.json","state":"https://pith.science/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IHETMTKKQU2X7CAJ7J5RZ2MGJO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:IHETMTKKQU2X7CAJ7J5RZ2MGJO","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":"f12af5e2252607aad1c4b8d0bcf2842b512fe3962979e1d83089c219002291ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-14T18:32:20Z","title_canon_sha256":"ecf08ca192cbe43b9ac2289dd42aad78cfc2f10533b99b94d728ffcd58424e07"},"schema_version":"1.0","source":{"id":"2312.09202","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2312.09202","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"arxiv_version","alias_value":"2312.09202v2","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.09202","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_12","alias_value":"IHETMTKKQU2X","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_16","alias_value":"IHETMTKKQU2X7CAJ","created_at":"2026-07-05T11:52:51Z"},{"alias_kind":"pith_short_8","alias_value":"IHETMTKK","created_at":"2026-07-05T11:52:51Z"}],"graph_snapshots":[{"event_id":"sha256:fccdb9bc84bd421bb7808cbd9f5138eca1a36cf0f0e3790ca37a91429805a22f","target":"graph","created_at":"2026-07-05T11:52:51Z","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/2312.09202/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.\n  Erd\\H{o}s, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite. Osthus, Pr\\\"omel, and Taraz later showed that for $m \\ge (1+\\epsilon)\\frac{\\sqrt{3}}{4}n^{3/2}\\sqrt{\\log n}$, almost every triangle-free graph on $n$ vertices and $m$ edges is bipartite. Here we give a precise characterization of the distribution of edges within ea","authors_text":"Aditya Potukuchi, Matthew Jenssen, Will Perkins","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-14T18:32:20Z","title":"On the evolution of structure in triangle-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.09202","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:a7b0477518b0d744d95f9e194c72149caeb16eafdb848253b925c0db11c724a4","target":"record","created_at":"2026-07-05T11:52:51Z","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":"f12af5e2252607aad1c4b8d0bcf2842b512fe3962979e1d83089c219002291ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-14T18:32:20Z","title_canon_sha256":"ecf08ca192cbe43b9ac2289dd42aad78cfc2f10533b99b94d728ffcd58424e07"},"schema_version":"1.0","source":{"id":"2312.09202","kind":"arxiv","version":2}},"canonical_sha256":"41c9364d4a85357f8809fa7b1ce9864baf2c093d6bab0c606846038aec8f70cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"41c9364d4a85357f8809fa7b1ce9864baf2c093d6bab0c606846038aec8f70cf","first_computed_at":"2026-07-05T11:52:51.711603Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:52:51.711603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XebpEPcHMNdsmLGiytJOJL3JHNRiKjTEW3BVjPaedO3VUcF7q/G2SoHIunmer2MxYuF3gZCQNBXP4p9fH7V/Aw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:52:51.712086Z","signed_message":"canonical_sha256_bytes"},"source_id":"2312.09202","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7b0477518b0d744d95f9e194c72149caeb16eafdb848253b925c0db11c724a4","sha256:fccdb9bc84bd421bb7808cbd9f5138eca1a36cf0f0e3790ca37a91429805a22f"],"state_sha256":"486c311ab9cce137527402bb933c9c176ee070c748ca1c622f5e69da447abd0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lNzlp/F2K3tLMqUB4TERzD9e/XIE68JnzNN50yYKUmUPbYCW1ZesNSgpG5QL/b/9Km6tau+W7IHTL5CWfj2BBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T18:18:25.575876Z","bundle_sha256":"2f2913dbbb26c2c03eb7bea5c35df5d4a262dbf8f5be1c1a379080d942d016d0"}}