{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:DZ4HIVPHLYFERYP63GGRLYLBYP","short_pith_number":"pith:DZ4HIVPH","schema_version":"1.0","canonical_sha256":"1e787455e75e0a48e1fed98d15e161c3f7bd48294ca813c0df39f4cbf37f00d0","source":{"kind":"arxiv","id":"2412.11901","version":2},"attestation_state":"computed","paper":{"title":"The Frankl-Pach upper bound is not tight for any uniformity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chi Hoi Yip, Gennian Ge, Shengtong Zhang, Xiaochen Zhao, Zixiang Xu","submitted_at":"2024-12-16T15:46:31Z","abstract_excerpt":"For any positive integers $n\\ge d+1\\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound $\\binom{n}{d}$ via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when $n$ is sufficiently large and $d$ is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires $d$ to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on"},"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":"2412.11901","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-12-16T15:46:31Z","cross_cats_sorted":[],"title_canon_sha256":"c70d2a9c4958fb3752492df156b77bfe89f08381179aab9d5690e08bc0a3bfaf","abstract_canon_sha256":"6288b61349a88b6c97a43e07e59a5888f54c480575564f64d9eea709891579eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:08:27.831674Z","signature_b64":"8CEdqME3BFxeLzwnixpaVewTJ2YDGIiIFHIoUxQBM+gwEWeikGvn1XbZyD3UEvUPRo66WD+4T7CUMgKACwUxCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e787455e75e0a48e1fed98d15e161c3f7bd48294ca813c0df39f4cbf37f00d0","last_reissued_at":"2026-06-09T02:08:27.830600Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:08:27.830600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Frankl-Pach upper bound is not tight for any uniformity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chi Hoi Yip, Gennian Ge, Shengtong Zhang, Xiaochen Zhao, Zixiang Xu","submitted_at":"2024-12-16T15:46:31Z","abstract_excerpt":"For any positive integers $n\\ge d+1\\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound $\\binom{n}{d}$ via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when $n$ is sufficiently large and $d$ is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires $d$ to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11901","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/2412.11901/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":"2412.11901","created_at":"2026-06-09T02:08:27.830753+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.11901v2","created_at":"2026-06-09T02:08:27.830753+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11901","created_at":"2026-06-09T02:08:27.830753+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZ4HIVPHLYFE","created_at":"2026-06-09T02:08:27.830753+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZ4HIVPHLYFERYP6","created_at":"2026-06-09T02:08:27.830753+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZ4HIVPH","created_at":"2026-06-09T02:08:27.830753+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.13850","citing_title":"Uniform set systems with small VC-dimension","ref_index":15,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP","json":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP.json","graph_json":"https://pith.science/api/pith-number/DZ4HIVPHLYFERYP63GGRLYLBYP/graph.json","events_json":"https://pith.science/api/pith-number/DZ4HIVPHLYFERYP63GGRLYLBYP/events.json","paper":"https://pith.science/paper/DZ4HIVPH"},"agent_actions":{"view_html":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP","download_json":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP.json","view_paper":"https://pith.science/paper/DZ4HIVPH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.11901&json=true","fetch_graph":"https://pith.science/api/pith-number/DZ4HIVPHLYFERYP63GGRLYLBYP/graph.json","fetch_events":"https://pith.science/api/pith-number/DZ4HIVPHLYFERYP63GGRLYLBYP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP/action/storage_attestation","attest_author":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP/action/author_attestation","sign_citation":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP/action/citation_signature","submit_replication":"https://pith.science/pith/DZ4HIVPHLYFERYP63GGRLYLBYP/action/replication_record"}},"created_at":"2026-06-09T02:08:27.830753+00:00","updated_at":"2026-06-09T02:08:27.830753+00:00"}