{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:Z26X2JSDQOMXMHJXTXV6WC373Y","short_pith_number":"pith:Z26X2JSD","canonical_record":{"source":{"id":"2407.01625","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-06-29T06:25:18Z","cross_cats_sorted":[],"title_canon_sha256":"4e0487bd24cce6512bee30d4c5a7ea24b433c5399616d0aea909c84de406ea74","abstract_canon_sha256":"e79f6bc56758169711d262170ebe9f4aa17f69ac45f6e379960aee090e1fe27a"},"schema_version":"1.0"},"canonical_sha256":"cebd7d26438399761d379debeb0b7fde06414175f26f0271d0464dd44eb61b56","source":{"kind":"arxiv","id":"2407.01625","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01625","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01625v2","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01625","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_12","alias_value":"Z26X2JSDQOMX","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_16","alias_value":"Z26X2JSDQOMXMHJX","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_8","alias_value":"Z26X2JSD","created_at":"2026-05-20T00:00:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:Z26X2JSDQOMXMHJXTXV6WC373Y","target":"record","payload":{"canonical_record":{"source":{"id":"2407.01625","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-06-29T06:25:18Z","cross_cats_sorted":[],"title_canon_sha256":"4e0487bd24cce6512bee30d4c5a7ea24b433c5399616d0aea909c84de406ea74","abstract_canon_sha256":"e79f6bc56758169711d262170ebe9f4aa17f69ac45f6e379960aee090e1fe27a"},"schema_version":"1.0"},"canonical_sha256":"cebd7d26438399761d379debeb0b7fde06414175f26f0271d0464dd44eb61b56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:15.707087Z","signature_b64":"EqEhdWzGKkOwyBvXkR+orL80L6vlufXWFQfGIMAbvrBr3sXvX80w94lbuguIjo6xBGWnEXABRQl0kjyTvtWgAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cebd7d26438399761d379debeb0b7fde06414175f26f0271d0464dd44eb61b56","last_reissued_at":"2026-05-20T00:00:15.706425Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:15.706425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.01625","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":"2026-05-20T00:00:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bxgrZcIVs2GKNYHpHqUbik/GphPYfDH0BbJnbkLTIlUf6raJW80BUr2/j/uLHN0mMT4JZwbO+pHGFigVnLQQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:18:36.305187Z"},"content_sha256":"2e27074adbb237a77d0c6479334224ed2b772c69d55e59ab0570eed084b95b56","schema_version":"1.0","event_id":"sha256:2e27074adbb237a77d0c6479334224ed2b772c69d55e59ab0570eed084b95b56"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:Z26X2JSDQOMXMHJXTXV6WC373Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Donglei Yang, Fan Yang, Jianfeng Hou, Yindong Jin","submitted_at":"2024-06-29T06:25:18Z","abstract_excerpt":"Let $ t\\ge s\\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\\Omega(d^{\\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times.\n  In 1975, Erd\\H{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite aver"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01625","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/2407.01625/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":"2026-05-20T00:00:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M4bSaUYpVje/IYBlmZQIJPYVlxI9leVhtk5BS3QMshlzIJe+EnWncZtAX9rs29x61xSjamTpfzCDtE1GdpCeDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:18:36.305918Z"},"content_sha256":"da92a98e2323de7a35d579374040fe293fb40d6d168a7c255e0123a91bf74146","schema_version":"1.0","event_id":"sha256:da92a98e2323de7a35d579374040fe293fb40d6d168a7c255e0123a91bf74146"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/bundle.json","state_url":"https://pith.science/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/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":"2026-05-27T20:18:36Z","links":{"resolver":"https://pith.science/pith/Z26X2JSDQOMXMHJXTXV6WC373Y","bundle":"https://pith.science/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/bundle.json","state":"https://pith.science/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z26X2JSDQOMXMHJXTXV6WC373Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:Z26X2JSDQOMXMHJXTXV6WC373Y","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":"e79f6bc56758169711d262170ebe9f4aa17f69ac45f6e379960aee090e1fe27a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-06-29T06:25:18Z","title_canon_sha256":"4e0487bd24cce6512bee30d4c5a7ea24b433c5399616d0aea909c84de406ea74"},"schema_version":"1.0","source":{"id":"2407.01625","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01625","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01625v2","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01625","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_12","alias_value":"Z26X2JSDQOMX","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_16","alias_value":"Z26X2JSDQOMXMHJX","created_at":"2026-05-20T00:00:15Z"},{"alias_kind":"pith_short_8","alias_value":"Z26X2JSD","created_at":"2026-05-20T00:00:15Z"}],"graph_snapshots":[{"event_id":"sha256:da92a98e2323de7a35d579374040fe293fb40d6d168a7c255e0123a91bf74146","target":"graph","created_at":"2026-05-20T00:00:15Z","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/2407.01625/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $ t\\ge s\\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\\Omega(d^{\\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times.\n  In 1975, Erd\\H{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite aver","authors_text":"Donglei Yang, Fan Yang, Jianfeng Hou, Yindong Jin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-06-29T06:25:18Z","title":"Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01625","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":"sha256:2e27074adbb237a77d0c6479334224ed2b772c69d55e59ab0570eed084b95b56","target":"record","created_at":"2026-05-20T00:00:15Z","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":"e79f6bc56758169711d262170ebe9f4aa17f69ac45f6e379960aee090e1fe27a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-06-29T06:25:18Z","title_canon_sha256":"4e0487bd24cce6512bee30d4c5a7ea24b433c5399616d0aea909c84de406ea74"},"schema_version":"1.0","source":{"id":"2407.01625","kind":"arxiv","version":2}},"canonical_sha256":"cebd7d26438399761d379debeb0b7fde06414175f26f0271d0464dd44eb61b56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cebd7d26438399761d379debeb0b7fde06414175f26f0271d0464dd44eb61b56","first_computed_at":"2026-05-20T00:00:15.706425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:15.706425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EqEhdWzGKkOwyBvXkR+orL80L6vlufXWFQfGIMAbvrBr3sXvX80w94lbuguIjo6xBGWnEXABRQl0kjyTvtWgAg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:15.707087Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.01625","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e27074adbb237a77d0c6479334224ed2b772c69d55e59ab0570eed084b95b56","sha256:da92a98e2323de7a35d579374040fe293fb40d6d168a7c255e0123a91bf74146"],"state_sha256":"c79c681b4f274791209eed921a718169a48aabd704306124c5eb3c0ee843e5d2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a5q4ipdmfoDedmXeoA49PEPA3VwvNkx4Vmy7fC4v7sCKKBjc9kWJiK4o6aPB8LE9nfEalMu8Kl6H82sLhQgKAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:18:36.309543Z","bundle_sha256":"dafd60d869763c2f7ec5c64cd9c748ce554ba3d61039ce7566a855c09f574033"}}