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Frankl and Nie proved a two-candidate formula for sufficiently large $n$ and conjectured it for all $n\\ge 2$; our formula shows that the conjectured expression must be enlarged, in small ranges of $n$, by additional Ahlswede--Khachatrian ball-type terms.\n  Our second result concern"},"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":"2606.06208","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-04T14:14:27Z","cross_cats_sorted":[],"title_canon_sha256":"560b86b5697d9cbbb00c9f1f90a49661c6bb69f979e39790c7258c84a5dd8463","abstract_canon_sha256":"9be188df66a945ff79e9d9c42b27c225696cacdab1c3bce221fdadeb66e15140"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:15:37.422850Z","signature_b64":"JtGWxmNxyr3nKAfANg5IXIIvmsE2JI3quyD2rVUmUqIJsEF5BxHJD7W05A7a+rM2lfsLBv4E8udi/KjitERBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a053b0a4ddfa3c2a811ab841f7339a0f18ca6af6258cbc2d8e05ab0957f153e","last_reissued_at":"2026-06-05T01:15:37.422352Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:15:37.422352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-trivial Intersection Problems for Multi-part Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caiyun Hu, Jianfeng Hou","submitted_at":"2026-06-04T14:14:27Z","abstract_excerpt":"We study non-trivial intersection problems for multi-part hypergraphs, excluding the usual extremal examples determined by fixed vertices or fixed coordinates. 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