{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:UIYVTPJK7WNHMNVZ5HAKFLDYLC","short_pith_number":"pith:UIYVTPJK","canonical_record":{"source":{"id":"2411.07090","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-11T16:12:09Z","cross_cats_sorted":[],"title_canon_sha256":"2e6542be431578bca266976e29a3a0ee8471361c2e57d367538f528f3f550245","abstract_canon_sha256":"004be0fd674ddffe78dc0d6e8ad2e5b29255579e990a79072aeeaa8bbbc759c6"},"schema_version":"1.0"},"canonical_sha256":"a23159bd2afd9a7636b9e9c0a2ac7858bde1301de2f5e25f2abbb819f4e7b640","source":{"kind":"arxiv","id":"2411.07090","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.07090","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"arxiv_version","alias_value":"2411.07090v2","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.07090","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_12","alias_value":"UIYVTPJK7WNH","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_16","alias_value":"UIYVTPJK7WNHMNVZ","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_8","alias_value":"UIYVTPJK","created_at":"2026-07-05T09:34:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:UIYVTPJK7WNHMNVZ5HAKFLDYLC","target":"record","payload":{"canonical_record":{"source":{"id":"2411.07090","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-11T16:12:09Z","cross_cats_sorted":[],"title_canon_sha256":"2e6542be431578bca266976e29a3a0ee8471361c2e57d367538f528f3f550245","abstract_canon_sha256":"004be0fd674ddffe78dc0d6e8ad2e5b29255579e990a79072aeeaa8bbbc759c6"},"schema_version":"1.0"},"canonical_sha256":"a23159bd2afd9a7636b9e9c0a2ac7858bde1301de2f5e25f2abbb819f4e7b640","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:34:22.967647Z","signature_b64":"Yuv+nGY7tBqRBNmkO9FvzUGYrRisZ3PQ5EY1mXzvZa0IMsA2H5YWWORgx7MZGvQRixLDzHv0qQr+5xDdwUwPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a23159bd2afd9a7636b9e9c0a2ac7858bde1301de2f5e25f2abbb819f4e7b640","last_reissued_at":"2026-07-05T09:34:22.967114Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:34:22.967114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.07090","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":"2026-07-05T09:34:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2RqsmWPxKRXl+yxeMLnS6cwrwPb+zVl+zjqEOR8JKC+ddu1zrTzdS/Y20V709h5zSWXD+QheDzvNiyh0t1USBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T10:57:39.239885Z"},"content_sha256":"d4110d50c6cf7ebe5582372692fc38b3ff59eedb371dda0dfad19f947886fc22","schema_version":"1.0","event_id":"sha256:d4110d50c6cf7ebe5582372692fc38b3ff59eedb371dda0dfad19f947886fc22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:UIYVTPJK7WNHMNVZ5HAKFLDYLC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive codegree Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s theorem for the generalized triangle","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Sijie Ren, Xizhi Liu","submitted_at":"2024-11-11T16:12:09Z","abstract_excerpt":"The celebrated Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite.\n  We establish a positive codegree extension of this result for the $r$-uniform generalized triangle $\\mathrm{T}_{r} = \\left\\{\\{1,\\ldots, r-1,r\\}, \\{1,\\ldots, r-1,r+1\\},\\{r,r+1, \\ldots, 2r-1\\}\\right\\}$$\\colon$ For every $n \\ge (r-1)(2r+1)/2$, if $\\mathcal{H}$ is an $n$-vertex $\\mathrm{T}_{r}$-free $r$-uniform hypergraph in which each $(r-1)$-tuple of vertices is contained in either zero edges or more than $2n/(2r+1)$ edges"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.07090","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/2411.07090/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":"2026-07-05T09:34:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IxD1DZlhRefSTIhgWmKEqaNQ56nuZo72wYWdABgGvnjMFYmUJYXo4slge1F1WoMOEEOY7VqRMLs8tdr3KnGnDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T10:57:39.240382Z"},"content_sha256":"71441100cb06037f3e6c96120ff00b0e567b881bb1cca8864be2376ef348ce4c","schema_version":"1.0","event_id":"sha256:71441100cb06037f3e6c96120ff00b0e567b881bb1cca8864be2376ef348ce4c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/bundle.json","state_url":"https://pith.science/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/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":"2026-08-05T10:57:39Z","links":{"resolver":"https://pith.science/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC","bundle":"https://pith.science/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/bundle.json","state":"https://pith.science/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UIYVTPJK7WNHMNVZ5HAKFLDYLC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:UIYVTPJK7WNHMNVZ5HAKFLDYLC","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":"004be0fd674ddffe78dc0d6e8ad2e5b29255579e990a79072aeeaa8bbbc759c6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-11T16:12:09Z","title_canon_sha256":"2e6542be431578bca266976e29a3a0ee8471361c2e57d367538f528f3f550245"},"schema_version":"1.0","source":{"id":"2411.07090","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.07090","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"arxiv_version","alias_value":"2411.07090v2","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.07090","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_12","alias_value":"UIYVTPJK7WNH","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_16","alias_value":"UIYVTPJK7WNHMNVZ","created_at":"2026-07-05T09:34:22Z"},{"alias_kind":"pith_short_8","alias_value":"UIYVTPJK","created_at":"2026-07-05T09:34:22Z"}],"graph_snapshots":[{"event_id":"sha256:71441100cb06037f3e6c96120ff00b0e567b881bb1cca8864be2376ef348ce4c","target":"graph","created_at":"2026-07-05T09:34:22Z","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/2411.07090/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The celebrated Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite.\n  We establish a positive codegree extension of this result for the $r$-uniform generalized triangle $\\mathrm{T}_{r} = \\left\\{\\{1,\\ldots, r-1,r\\}, \\{1,\\ldots, r-1,r+1\\},\\{r,r+1, \\ldots, 2r-1\\}\\right\\}$$\\colon$ For every $n \\ge (r-1)(2r+1)/2$, if $\\mathcal{H}$ is an $n$-vertex $\\mathrm{T}_{r}$-free $r$-uniform hypergraph in which each $(r-1)$-tuple of vertices is contained in either zero edges or more than $2n/(2r+1)$ edges","authors_text":"Jian Wang, Sijie Ren, Xizhi Liu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-11T16:12:09Z","title":"Positive codegree Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s theorem for the generalized triangle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.07090","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:d4110d50c6cf7ebe5582372692fc38b3ff59eedb371dda0dfad19f947886fc22","target":"record","created_at":"2026-07-05T09:34:22Z","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":"004be0fd674ddffe78dc0d6e8ad2e5b29255579e990a79072aeeaa8bbbc759c6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-11T16:12:09Z","title_canon_sha256":"2e6542be431578bca266976e29a3a0ee8471361c2e57d367538f528f3f550245"},"schema_version":"1.0","source":{"id":"2411.07090","kind":"arxiv","version":2}},"canonical_sha256":"a23159bd2afd9a7636b9e9c0a2ac7858bde1301de2f5e25f2abbb819f4e7b640","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a23159bd2afd9a7636b9e9c0a2ac7858bde1301de2f5e25f2abbb819f4e7b640","first_computed_at":"2026-07-05T09:34:22.967114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:34:22.967114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Yuv+nGY7tBqRBNmkO9FvzUGYrRisZ3PQ5EY1mXzvZa0IMsA2H5YWWORgx7MZGvQRixLDzHv0qQr+5xDdwUwPCw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:34:22.967647Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.07090","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4110d50c6cf7ebe5582372692fc38b3ff59eedb371dda0dfad19f947886fc22","sha256:71441100cb06037f3e6c96120ff00b0e567b881bb1cca8864be2376ef348ce4c"],"state_sha256":"91db8a5ac1cfe4c6942444b19ac2ba183bb8ac422dc15b433adf082747271001"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iRxTdbi635bBpbJ7TyQRtDsCNng0dzd95hWOykV6cS9ZCOlvMff57P4ikBl9oJNZ38u4kyyznbsfyegD6Am6Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T10:57:39.244813Z","bundle_sha256":"4c1bcaec40cea21c360f682b9c056ea3f6b8f51033b24baf9a59728720abedd4"}}