{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:DVQZ4YCKYYXXHFYCPR2L666M2U","short_pith_number":"pith:DVQZ4YCK","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"},"canonical_sha256":"1d619e604ac62f7397027c74bf7bccd5312037c7b1d08d66b9e16ab604f8a0ff","source":{"kind":"arxiv","id":"1203.0563","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:DVQZ4YCKYYXXHFYCPR2L666M2U","target":"record","payload":{"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"},"canonical_sha256":"1d619e604ac62f7397027c74bf7bccd5312037c7b1d08d66b9e16ab604f8a0ff","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"},"source_kind":"arxiv","source_id":"1203.0563","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ER6qsKtNFfHABxVbfrR5oAhts5h1ydidSUUs5rkkG3EhLEu466S1SvYlNMTpfzsBbFrxP9DwbPjdEU+nO8HtCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:00:17.091041Z"},"content_sha256":"92022a24648b87ff7cd75a3dbae4f3b4ba204abe97325e509fc5fca404c37b10","schema_version":"1.0","event_id":"sha256:92022a24648b87ff7cd75a3dbae4f3b4ba204abe97325e509fc5fca404c37b10"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:DVQZ4YCKYYXXHFYCPR2L666M2U","target":"graph","payload":{"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 $D(n)=\\lceil n/2 \\rceil$. Here we show that $D(n) \\geq n/2 + n/236 - O(\\sqrt{n})$ and thereby disprove this conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0563","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rxzj4Nk5zPUKzsue4Ya0F0OEZzayGRSdolTBjJixQGSjezq3qoU5lZGROuf4uQvURfeCSQmf3dOD4qWIXQ7ABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:00:17.091661Z"},"content_sha256":"2ee7a028b1e08ca0095638cfbf5dcb75dfc57060524739a3233b281077efe630","schema_version":"1.0","event_id":"sha256:2ee7a028b1e08ca0095638cfbf5dcb75dfc57060524739a3233b281077efe630"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/bundle.json","state_url":"https://pith.science/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T13:00:17Z","links":{"resolver":"https://pith.science/pith/DVQZ4YCKYYXXHFYCPR2L666M2U","bundle":"https://pith.science/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/bundle.json","state":"https://pith.science/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DVQZ4YCKYYXXHFYCPR2L666M2U/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sLA5fViLHpIhZmvvt3mDdRwtfmDJDtzPm0sd8r7BWfKlWRDkJcg8T9r4D139nOb8pcuJEVaCVUxZyMMYi9qfCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:00:17.094870Z","bundle_sha256":"ea96d22204caa1e202810417fc8dd97f79a0cccf556234277f01e460a867d9e5"}}