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Given a number $0<r\\le 1$, a set $S\\subset \\mathbb{R}^2$ is called $r$-fat if there exists a point $c\\in S$ such that $B(c,r) \\subseteq S\\subseteq B(c,1)$, where $B(c,r)\\subset \\mathbb{R}^2$ is a disk of radius $r$ with center-point $c$. We prove constant upper bounds $C=C(r)$ on the piercing numbers in families of $r$-fat sets in $\\mathbb{R}^2$ that satisfy the $(2,2)$ or the $(4,3)$ properties. This extends results by Danzer and Karasev on the piercing numbers in intersecting families of disk"},"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":"1711.05308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-14T20:45:19Z","cross_cats_sorted":[],"title_canon_sha256":"1ed1d3ae4ab59330f02a3d1ca5d84ed76212b40eb48fb5e8d17338bd8ddd175f","abstract_canon_sha256":"47358ae14afca04dd805318a3f953e72acaf34d4d9afd824226286c4f383aac1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:33.686101Z","signature_b64":"XtkPeDoQ5gfprp1S6Kmu4ug2+uGz/QgA4PKEOqtiNTTm5LE41QeflohKRw+kTTMvJasxwHT0LdGhEsAnEkD3AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd59cefa8f0f11f4cbf5f18e6b37b94dca864f5075aab5ae05f614f79b249458","last_reissued_at":"2026-05-18T00:30:33.685496Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:33.685496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $(2,2)$ and $(4,3)$ properties in families of fat sets in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shiliang Gao, Shira Zerbib","submitted_at":"2017-11-14T20:45:19Z","abstract_excerpt":"A family of sets satisfies the $(p,q)$ property if among every $p$ members of it some $q$ intersect. 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