{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QNGORMMF2D4AIV4K757ZAWVBCW","short_pith_number":"pith:QNGORMMF","canonical_record":{"source":{"id":"2607.04323","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T14:14:28Z","cross_cats_sorted":[],"title_canon_sha256":"888dededa2603272d6002caef0c80b51e9cef23bf8cf06551714a5269835e3f5","abstract_canon_sha256":"fca80e27c5719267a1d8d41cb4cb707c571ea1d2798f2b4a2b8368c83072808f"},"schema_version":"1.0"},"canonical_sha256":"834ce8b185d0f804578aff7f905aa115a79b33d0b595c5d0f07d866e7fe8cf0d","source":{"kind":"arxiv","id":"2607.04323","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04323","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04323v1","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04323","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_12","alias_value":"QNGORMMF2D4A","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_16","alias_value":"QNGORMMF2D4AIV4K","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_8","alias_value":"QNGORMMF","created_at":"2026-07-07T02:19:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QNGORMMF2D4AIV4K757ZAWVBCW","target":"record","payload":{"canonical_record":{"source":{"id":"2607.04323","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T14:14:28Z","cross_cats_sorted":[],"title_canon_sha256":"888dededa2603272d6002caef0c80b51e9cef23bf8cf06551714a5269835e3f5","abstract_canon_sha256":"fca80e27c5719267a1d8d41cb4cb707c571ea1d2798f2b4a2b8368c83072808f"},"schema_version":"1.0"},"canonical_sha256":"834ce8b185d0f804578aff7f905aa115a79b33d0b595c5d0f07d866e7fe8cf0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:08.771228Z","signature_b64":"VLybJXYZwdSQnOrP2htsf7zatGWf7s/LQXBvM80q8RUhCc3O8Gg8t0e9TPJTiifII1HC8VOQIYRaiQz8A+HHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"834ce8b185d0f804578aff7f905aa115a79b33d0b595c5d0f07d866e7fe8cf0d","last_reissued_at":"2026-07-07T02:19:08.770309Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:08.770309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.04323","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-07T02:19:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+zqMfl+Q7IVMhUdUXhybf3AuK7Re9vcSdjf2aNgJZvIDDjRzX+pLTseUjZSO09KRDmyk6kJ4c2hKZ/IGXIa4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T02:41:23.548740Z"},"content_sha256":"763ecaf743bc36c04a7a3391e002699197222a60d0ac4cabda576157a4464d15","schema_version":"1.0","event_id":"sha256:763ecaf743bc36c04a7a3391e002699197222a60d0ac4cabda576157a4464d15"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QNGORMMF2D4AIV4K757ZAWVBCW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the structure of dense graphs with given odd girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shipeng Wang, Xingyan Lu","submitted_at":"2026-07-05T14:14:28Z","abstract_excerpt":"A classical theorem of Andr\\'asfai, Erd\\H{o}s, and S\\'os states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\\delta(G)>\\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$.\n  In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{4n}{6k-1}$ is homomorphic to the M\\\"obius ladder on $4k$ vertices. This answers a question of M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04323","kind":"arxiv","version":1},"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/2607.04323/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":"2026-07-07T02:19:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dyMPOsc32XP70NqFEHhyOvm/T/6VKsVrTRaxqB4fMRV80FGiG2VyJFEz+x1e5/IBKj6y1dPmjxjYPLFkNoH5CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T02:41:23.549272Z"},"content_sha256":"6b0ab6752708459e545d4e73bedf5fe8f14491ee0e485c4b34eda14aef9fd0e9","schema_version":"1.0","event_id":"sha256:6b0ab6752708459e545d4e73bedf5fe8f14491ee0e485c4b34eda14aef9fd0e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QNGORMMF2D4AIV4K757ZAWVBCW/bundle.json","state_url":"https://pith.science/pith/QNGORMMF2D4AIV4K757ZAWVBCW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QNGORMMF2D4AIV4K757ZAWVBCW/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":"2026-08-06T02:41:23Z","links":{"resolver":"https://pith.science/pith/QNGORMMF2D4AIV4K757ZAWVBCW","bundle":"https://pith.science/pith/QNGORMMF2D4AIV4K757ZAWVBCW/bundle.json","state":"https://pith.science/pith/QNGORMMF2D4AIV4K757ZAWVBCW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QNGORMMF2D4AIV4K757ZAWVBCW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QNGORMMF2D4AIV4K757ZAWVBCW","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":"fca80e27c5719267a1d8d41cb4cb707c571ea1d2798f2b4a2b8368c83072808f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T14:14:28Z","title_canon_sha256":"888dededa2603272d6002caef0c80b51e9cef23bf8cf06551714a5269835e3f5"},"schema_version":"1.0","source":{"id":"2607.04323","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04323","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04323v1","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04323","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_12","alias_value":"QNGORMMF2D4A","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_16","alias_value":"QNGORMMF2D4AIV4K","created_at":"2026-07-07T02:19:08Z"},{"alias_kind":"pith_short_8","alias_value":"QNGORMMF","created_at":"2026-07-07T02:19:08Z"}],"graph_snapshots":[{"event_id":"sha256:6b0ab6752708459e545d4e73bedf5fe8f14491ee0e485c4b34eda14aef9fd0e9","target":"graph","created_at":"2026-07-07T02:19:08Z","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/2607.04323/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A classical theorem of Andr\\'asfai, Erd\\H{o}s, and S\\'os states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\\delta(G)>\\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$.\n  In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{4n}{6k-1}$ is homomorphic to the M\\\"obius ladder on $4k$ vertices. This answers a question of M","authors_text":"Shipeng Wang, Xingyan Lu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T14:14:28Z","title":"On the structure of dense graphs with given odd girth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04323","kind":"arxiv","version":1},"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":"sha256:763ecaf743bc36c04a7a3391e002699197222a60d0ac4cabda576157a4464d15","target":"record","created_at":"2026-07-07T02:19:08Z","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":"fca80e27c5719267a1d8d41cb4cb707c571ea1d2798f2b4a2b8368c83072808f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T14:14:28Z","title_canon_sha256":"888dededa2603272d6002caef0c80b51e9cef23bf8cf06551714a5269835e3f5"},"schema_version":"1.0","source":{"id":"2607.04323","kind":"arxiv","version":1}},"canonical_sha256":"834ce8b185d0f804578aff7f905aa115a79b33d0b595c5d0f07d866e7fe8cf0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"834ce8b185d0f804578aff7f905aa115a79b33d0b595c5d0f07d866e7fe8cf0d","first_computed_at":"2026-07-07T02:19:08.770309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:19:08.770309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VLybJXYZwdSQnOrP2htsf7zatGWf7s/LQXBvM80q8RUhCc3O8Gg8t0e9TPJTiifII1HC8VOQIYRaiQz8A+HHDw==","signature_status":"signed_v1","signed_at":"2026-07-07T02:19:08.771228Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.04323","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:763ecaf743bc36c04a7a3391e002699197222a60d0ac4cabda576157a4464d15","sha256:6b0ab6752708459e545d4e73bedf5fe8f14491ee0e485c4b34eda14aef9fd0e9"],"state_sha256":"7331d323fda3af36b868e35275255c5c0b8a8eef1868c265094b60cc531c8571"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7mwnBJcsVZdnhV9ZZZXl1xmACIXE8yEVxzwPXvcT/djjhhxog9dL94unMx21PbCQbZGAeHl6uC/Z6UMka35hBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T02:41:23.554124Z","bundle_sha256":"970c4c2e3713f8a519f9c6d5718b91bbcf8091720669ee1b671c9e72b1d907e6"}}