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Frankl conjectured that for $n>2k$ and every intersecting family $\\mathcal F\\subseteq [n]^{(k)}$, there is some $i\\in[n]$ such that $\\vert \\partial \\mathcal F(i)\\vert\\geq \\vert\\mathcal F(i)\\vert$, where $\\mathcal F(i)=\\{F\\setminus i:i\\in F\\in\\mathcal F\\}$ is the link of $\\mathcal F$ at $i$. Here, we prove this conjecture in a very strong "},"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":"2206.04278","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-06-09T04:57:01Z","cross_cats_sorted":[],"title_canon_sha256":"886a435a79eac0bc34e876a5bcaf50cce8b4d08bdae236a081bdca41fee9db5b","abstract_canon_sha256":"3a12472700380975feb268b10dfc355a09a3753e508f187251e1c375bfbbc8c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:30:25.069551Z","signature_b64":"K7JZfaEBn57Uy4LRIjGy5FGbriFGY89Omaq/brEf90yV8TsI9KCXn2eidJ9vfHh6GE7IOdEZsgLPq3NuKLLEDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0489cad76a74068f76da3c12693659536d148c5a548eeb6c956f1e5f03c4fae","last_reissued_at":"2026-07-05T04:30:25.069098Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:30:25.069098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A local version of Katona's intersection theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Sch\\\"ulke, Marcelo Sales","submitted_at":"2022-06-09T04:57:01Z","abstract_excerpt":"Katona's intersection theorem states that every intersecting family $\\mathcal F\\subseteq[n]^{(k)}$ satisfies $\\vert\\partial\\mathcal F\\vert\\geq\\vert\\mathcal F\\vert$, where $\\partial\\mathcal F=\\{F\\setminus x:x\\in F\\in\\mathcal F\\}$ is the shadow of $\\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\\mathcal F\\subseteq [n]^{(k)}$, there is some $i\\in[n]$ such that $\\vert \\partial \\mathcal F(i)\\vert\\geq \\vert\\mathcal F(i)\\vert$, where $\\mathcal F(i)=\\{F\\setminus i:i\\in F\\in\\mathcal F\\}$ is the link of $\\mathcal F$ at $i$. 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