{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:O3JNPD65RIIUWVEXAAAQ3COLM7","short_pith_number":"pith:O3JNPD65","schema_version":"1.0","canonical_sha256":"76d2d78fdd8a114b549700010d89cb67f4253a1842ba6153fa75c2d16f6b5962","source":{"kind":"arxiv","id":"2606.22064","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.22064","created_at":"2026-06-23T02:13:06.781918+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.22064v1","created_at":"2026-06-23T02:13:06.781918+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22064","created_at":"2026-06-23T02:13:06.781918+00:00"},{"alias_kind":"pith_short_12","alias_value":"O3JNPD65RIIU","created_at":"2026-06-23T02:13:06.781918+00:00"},{"alias_kind":"pith_short_16","alias_value":"O3JNPD65RIIUWVEX","created_at":"2026-06-23T02:13:06.781918+00:00"},{"alias_kind":"pith_short_8","alias_value":"O3JNPD65","created_at":"2026-06-23T02:13:06.781918+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2606.24776","citing_title":"A disproof of the uniform witness conjecture","ref_index":21,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7","json":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7.json","graph_json":"https://pith.science/api/pith-number/O3JNPD65RIIUWVEXAAAQ3COLM7/graph.json","events_json":"https://pith.science/api/pith-number/O3JNPD65RIIUWVEXAAAQ3COLM7/events.json","paper":"https://pith.science/paper/O3JNPD65"},"agent_actions":{"view_html":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7","download_json":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7.json","view_paper":"https://pith.science/paper/O3JNPD65","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.22064&json=true","fetch_graph":"https://pith.science/api/pith-number/O3JNPD65RIIUWVEXAAAQ3COLM7/graph.json","fetch_events":"https://pith.science/api/pith-number/O3JNPD65RIIUWVEXAAAQ3COLM7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/action/storage_attestation","attest_author":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/action/author_attestation","sign_citation":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/action/citation_signature","submit_replication":"https://pith.science/pith/O3JNPD65RIIUWVEXAAAQ3COLM7/action/replication_record"}},"created_at":"2026-06-23T02:13:06.781918+00:00","updated_at":"2026-06-23T02:13:06.781918+00:00"}