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We prove that if each $k$-wise and $(k+1)$-wise intersection of sets from $\\mathcal A$ is empty, or a single point, or both open and path-connected, then $f_{k+1}=0$ implies $f_k \\leq cf_{k-1}$ for some positive constant $c$ depending only on $k$. Similarly, let $b \\geq 2, k > 2b$ be integers. We prove that if each $k$-wise or $(k+1"},"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":"1907.00885","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-01T15:59:05Z","cross_cats_sorted":[],"title_canon_sha256":"c4c00ff5a8130a8530fc2316722f0fa3a66dd0233bed337da19488f59aa89ebc","abstract_canon_sha256":"42c046359437f8b0dfb55c194c56f0946fec5cd6e61ff8437d7623b6958dbe42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:26:08.826281Z","signature_b64":"3Z5fDV412Gl8s514cfb1wNWZaQEW3/7xut2jlNkj5fYEPn5g9qN6GiSBum7+liFo7e6cow920uiXqZMSCHKaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b6b67d388e10bf1dd397b3303c10ea6d917c1a013a1de7e9804859fecf53914","last_reissued_at":"2026-07-05T00:26:08.825850Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:26:08.825850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Intersection patterns of planar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gil Kalai, Zuzana Pat\\'akov\\'a","submitted_at":"2019-07-01T15:59:05Z","abstract_excerpt":"Let $\\mathcal A=\\{A_1,\\ldots,A_n\\}$ be a family of sets in the plane. 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