{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:O3JNPD65RIIUWVEXAAAQ3COLM7","short_pith_number":"pith:O3JNPD65","canonical_record":{"source":{"id":"2606.22064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T14:26:23Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"7db93955649dba89b2e11fb0bd77480d01ac903905cb00da9cb56668c7d391d7","abstract_canon_sha256":"eb93b3b61df660017c474d29531ac8b13d9f2a70c0cc515eaf250c1b52526122"},"schema_version":"1.0"},"canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","source":{"kind":"arxiv","id":"2606.22064","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22064","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22064v1","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22064","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_12","alias_value":"O3JNPD65RIIU","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_16","alias_value":"O3JNPD65RIIUWVEX","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_8","alias_value":"O3JNPD65","created_at":"2026-06-23T02:13:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:O3JNPD65RIIUWVEXAAAQ3COLM7","target":"record","payload":{"canonical_record":{"source":{"id":"2606.22064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T14:26:23Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"7db93955649dba89b2e11fb0bd77480d01ac903905cb00da9cb56668c7d391d7","abstract_canon_sha256":"eb93b3b61df660017c474d29531ac8b13d9f2a70c0cc515eaf250c1b52526122"},"schema_version":"1.0"},"canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:06.782263Z","signature_b64":"vCEi7+hP2kKgB1n36pGWWO1O/AMJAWB19LQGSjLLwxHkatOLQcRQCa8XzsmYWShRL3mdPeb8PmIVi8vTHWs7CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","last_reissued_at":"2026-06-23T02:13:06.781854Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:06.781854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.22064","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":"2026-06-23T02:13:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yw9eJpmWGAXP9jIPkqJC7wBd/PRoGVppRXNq739Jp/8Q/8SRc14b3zbzSXfVeEpn8ja5ILDT6/OtheVotfGTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T13:44:09.516158Z"},"content_sha256":"aa180e84e962a0fe203affae20a36183e8069570e36baf1374ecbad60676f6b7","schema_version":"1.0","event_id":"sha256:aa180e84e962a0fe203affae20a36183e8069570e36baf1374ecbad60676f6b7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:O3JNPD65RIIUWVEXAAAQ3COLM7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recursive lower bounds for uniform set systems of bounded VC-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.CO","authors_text":"Gennian Ge, Xiaochen Zhao","submitted_at":"2026-06-20T14:26:23Z","abstract_excerpt":"For integers $n\\ge d+1$, let $\\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \\[\n  \\mathsf{M}_d(n)\\ge \\binom{n-1}{d}+\\binom{n-4}{d-2}. \\] We introduce a two-cover lifting construction and prove the recursive lower bound \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4}{d-2}+\\mathsf{M}_{d-3}(n-5) \\] for every $d\\ge 3$ and $n\\ge d+3$. Consequently, \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22064","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/2606.22064/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":"2026-06-23T02:13:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vPONVjtZatnULqJjhCUz9IO5QKBknc0rJX4u6gnRN9a9xIWSVLdqXdqiB+OITaueAO5jY2WB6pOFn8EFif/8BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T13:44:09.516724Z"},"content_sha256":"8686ea71887ce288d23932a5edc02de74147bc3774aeab54ffa0b5d55cc9f043","schema_version":"1.0","event_id":"sha256:8686ea71887ce288d23932a5edc02de74147bc3774aeab54ffa0b5d55cc9f043"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/bundle.json","state_url":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/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":"2026-08-05T13:44:09Z","links":{"resolver":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7","bundle":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/bundle.json","state":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:O3JNPD65RIIUWVEXAAAQ3COLM7","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":"eb93b3b61df660017c474d29531ac8b13d9f2a70c0cc515eaf250c1b52526122","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T14:26:23Z","title_canon_sha256":"7db93955649dba89b2e11fb0bd77480d01ac903905cb00da9cb56668c7d391d7"},"schema_version":"1.0","source":{"id":"2606.22064","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22064","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22064v1","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22064","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_12","alias_value":"O3JNPD65RIIU","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_16","alias_value":"O3JNPD65RIIUWVEX","created_at":"2026-06-23T02:13:06Z"},{"alias_kind":"pith_short_8","alias_value":"O3JNPD65","created_at":"2026-06-23T02:13:06Z"}],"graph_snapshots":[{"event_id":"sha256:8686ea71887ce288d23932a5edc02de74147bc3774aeab54ffa0b5d55cc9f043","target":"graph","created_at":"2026-06-23T02:13:06Z","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/2606.22064/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For integers $n\\ge d+1$, let $\\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \\[\n  \\mathsf{M}_d(n)\\ge \\binom{n-1}{d}+\\binom{n-4}{d-2}. \\] We introduce a two-cover lifting construction and prove the recursive lower bound \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4}{d-2}+\\mathsf{M}_{d-3}(n-5) \\] for every $d\\ge 3$ and $n\\ge d+3$. Consequently, \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4","authors_text":"Gennian Ge, Xiaochen Zhao","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T14:26:23Z","title":"Recursive lower bounds for uniform set systems of bounded VC-dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22064","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:aa180e84e962a0fe203affae20a36183e8069570e36baf1374ecbad60676f6b7","target":"record","created_at":"2026-06-23T02:13:06Z","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":"eb93b3b61df660017c474d29531ac8b13d9f2a70c0cc515eaf250c1b52526122","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T14:26:23Z","title_canon_sha256":"7db93955649dba89b2e11fb0bd77480d01ac903905cb00da9cb56668c7d391d7"},"schema_version":"1.0","source":{"id":"2606.22064","kind":"arxiv","version":1}},"canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","first_computed_at":"2026-06-23T02:13:06.781854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T02:13:06.781854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vCEi7+hP2kKgB1n36pGWWO1O/AMJAWB19LQGSjLLwxHkatOLQcRQCa8XzsmYWShRL3mdPeb8PmIVi8vTHWs7CA==","signature_status":"signed_v1","signed_at":"2026-06-23T02:13:06.782263Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.22064","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa180e84e962a0fe203affae20a36183e8069570e36baf1374ecbad60676f6b7","sha256:8686ea71887ce288d23932a5edc02de74147bc3774aeab54ffa0b5d55cc9f043"],"state_sha256":"1f75b951f3aca33a08c6c529dd5d6edf58b1894fb9a2f2fe7c76a950123302d1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pQxV2fIEr8M5LUpvbrhSoEwOdhQfZtCIbi4NLkCVMmIavg5sXsvGxQf50JXlGurMOPfQ+AFLSRC98z/Yer1dAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T13:44:09.520922Z","bundle_sha256":"1b6ec1d833171413e52ef5061282eb24369c6a58a7519463329ae6143ad92d69"}}