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Here we show that $D(n) \\geq n/2 + n/236 - O(\\sqrt{n})$ and thereby disprove this conjecture."},"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":"1203.0563","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-02T20:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"1947f1751c38153b735d743d02792534497c0791bbefee86dbde23112c982a00","abstract_canon_sha256":"a29fbb8f31a5d920011054f0b44b9c69520580b8f2da929b3d07bdc8bf7568c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:58.115815Z","signature_b64":"aRMTUJfENg+XYElD7+iJRNU2BHebupFe+OHuiSICx+CMJL7FVCYlMA2XSAhIwFudfUilBtmSAsVIbycoZ3VxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d619e604ac62f7397027c74bf7bccd5312037c7b1d08d66b9e16ab604f8a0ff","last_reissued_at":"2026-05-18T03:49:58.115004Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:58.115004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Disjoint empty disks supported by a point set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adrian Dumitrescu, Minghui Jiang","submitted_at":"2012-03-02T20:49:21Z","abstract_excerpt":"For a planar point-set $P$, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in $P$ lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and Urabe recently conjectured that $D(n)=\\lceil n/2 \\rceil$. 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