{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:DVQZ4YCKYYXXHFYCPR2L666M2U","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a29fbb8f31a5d920011054f0b44b9c69520580b8f2da929b3d07bdc8bf7568c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-02T20:49:21Z","title_canon_sha256":"1947f1751c38153b735d743d02792534497c0791bbefee86dbde23112c982a00"},"schema_version":"1.0","source":{"id":"1203.0563","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.0563","created_at":"2026-05-18T03:49:58Z"},{"alias_kind":"arxiv_version","alias_value":"1203.0563v2","created_at":"2026-05-18T03:49:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0563","created_at":"2026-05-18T03:49:58Z"},{"alias_kind":"pith_short_12","alias_value":"DVQZ4YCKYYXX","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DVQZ4YCKYYXXHFYC","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DVQZ4YCK","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:2ee7a028b1e08ca0095638cfbf5dcb75dfc57060524739a3233b281077efe630","target":"graph","created_at":"2026-05-18T03:49:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $D(n)=\\lceil n/2 \\rceil$. Here we show that $D(n) \\geq n/2 + n/236 - O(\\sqrt{n})$ and thereby disprove this conjecture.","authors_text":"Adrian Dumitrescu, Minghui Jiang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-02T20:49:21Z","title":"Disjoint empty disks supported by a point set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0563","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:92022a24648b87ff7cd75a3dbae4f3b4ba204abe97325e509fc5fca404c37b10","target":"record","created_at":"2026-05-18T03:49:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a29fbb8f31a5d920011054f0b44b9c69520580b8f2da929b3d07bdc8bf7568c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-02T20:49:21Z","title_canon_sha256":"1947f1751c38153b735d743d02792534497c0791bbefee86dbde23112c982a00"},"schema_version":"1.0","source":{"id":"1203.0563","kind":"arxiv","version":2}},"canonical_sha256":"1d619e604ac62f7397027c74bf7bccd5312037c7b1d08d66b9e16ab604f8a0ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d619e604ac62f7397027c74bf7bccd5312037c7b1d08d66b9e16ab604f8a0ff","first_computed_at":"2026-05-18T03:49:58.115004Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:58.115004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aRMTUJfENg+XYElD7+iJRNU2BHebupFe+OHuiSICx+CMJL7FVCYlMA2XSAhIwFudfUilBtmSAsVIbycoZ3VxBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:58.115815Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.0563","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92022a24648b87ff7cd75a3dbae4f3b4ba204abe97325e509fc5fca404c37b10","sha256:2ee7a028b1e08ca0095638cfbf5dcb75dfc57060524739a3233b281077efe630"],"state_sha256":"077610081cff9b49b80fab6846877c01d888204e708c09defadd555c2ecaafa4"}