{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QK5KLLWTDVBZWZP4723NZRBTSN","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":"156c9e3fb3137bb59859ee0109ca402dea2ebc969c2acabf3429d7264c8066e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-02T15:11:25Z","title_canon_sha256":"8d480ec5b7e77765495ecddaa6032fd13bbf6fe57b58da1f8aa577d4e049249f"},"schema_version":"1.0","source":{"id":"1411.0266","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.0266","created_at":"2026-05-18T02:38:52Z"},{"alias_kind":"arxiv_version","alias_value":"1411.0266v1","created_at":"2026-05-18T02:38:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0266","created_at":"2026-05-18T02:38:52Z"},{"alias_kind":"pith_short_12","alias_value":"QK5KLLWTDVBZ","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QK5KLLWTDVBZWZP4","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QK5KLLWT","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:a9a595b601f73f0ccd76b9642edc443ce5afe40859170978b8fa0f21e231cf06","target":"graph","created_at":"2026-05-18T02:38:52Z","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":"In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n;m, where m ? n, let S(n;m) be the smallest integer s for which there exists a PBD on n points whose largest block has size m and the sum of its block sizes is equal to s. Also, let S0(n;m) be the smallest integers for which there exists a PBD on n points which has a block of size m and the sum of it block sizes is equal to s. We prove some lower bounds for S(n;m) and S0(n;m)","authors_text":"Akbar Davoodi, Behnaz Omoomi, Ramin Javadi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-02T15:11:25Z","title":"Pairwise Balanced Designs and Sigma Clique Partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0266","kind":"arxiv","version":1},"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":"sha256:18189290a420293f023e87e43afe79b786692be144a7bd4001be130475fc5589","target":"record","created_at":"2026-05-18T02:38:52Z","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":"156c9e3fb3137bb59859ee0109ca402dea2ebc969c2acabf3429d7264c8066e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-02T15:11:25Z","title_canon_sha256":"8d480ec5b7e77765495ecddaa6032fd13bbf6fe57b58da1f8aa577d4e049249f"},"schema_version":"1.0","source":{"id":"1411.0266","kind":"arxiv","version":1}},"canonical_sha256":"82baa5aed31d439b65fcfeb6dcc433934d1ca326c5fc75cd1ee7fc0da03cdc66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82baa5aed31d439b65fcfeb6dcc433934d1ca326c5fc75cd1ee7fc0da03cdc66","first_computed_at":"2026-05-18T02:38:52.116222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:52.116222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T/tmXicV2HcgpdZgqqDTrHqzZ8k3kRT9lWz1uKvyRX1IFswxZIGwfgC+YN1KiU227T4rQemf9dc4WY3FNwdYCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:52.116672Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.0266","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18189290a420293f023e87e43afe79b786692be144a7bd4001be130475fc5589","sha256:a9a595b601f73f0ccd76b9642edc443ce5afe40859170978b8fa0f21e231cf06"],"state_sha256":"5c331b47ce6ae5f85b84da1acbebe8134cb11a64513586de5a1027a739e7fb8a"}