{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:GGQPKITEFHNNYZOYUDHMIY6POA","short_pith_number":"pith:GGQPKITE","schema_version":"1.0","canonical_sha256":"31a0f5226429dadc65d8a0cec463cf703237ab1adc1de316409852884cc7cb6f","source":{"kind":"arxiv","id":"1209.3598","version":2},"attestation_state":"computed","paper":{"title":"Exact Bounds for Some Hypergraph Saturation Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Asaf Shapira, Guy Moshkovitz","submitted_at":"2012-09-17T09:13:10Z","abstract_excerpt":"Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a new edge is added, a new copy of K_{p,q} is created. The problem of bounding W_n(p,q), and its natural hypergraph generalization, was introduced by Balogh, Bollob\\'as, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W_n(1,q).\n  Our main results in this paper give exact bounds for W_n(p,q), its hypergraph analog"},"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":"1209.3598","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T09:13:10Z","cross_cats_sorted":[],"title_canon_sha256":"a9fba7e3e2372111d308360c76411dd2ecf5b7f4fa2c2483c4cc864377268ed4","abstract_canon_sha256":"5fbf17a70f8451c5076ee73428913c01af8fdbeae7e7ea99dbeb771866479416"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:36.005876Z","signature_b64":"syn8L2yQyM2hyKkVbSIlypLoJCZhOkDU7NAhY+KRkUT5xKMDjOIgKjKFe/vMGzTIUXL4Dkn9xqzqoAJxrAsEDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31a0f5226429dadc65d8a0cec463cf703237ab1adc1de316409852884cc7cb6f","last_reissued_at":"2026-05-18T03:01:36.005219Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:36.005219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact Bounds for Some Hypergraph Saturation Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Asaf Shapira, Guy Moshkovitz","submitted_at":"2012-09-17T09:13:10Z","abstract_excerpt":"Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a new edge is added, a new copy of K_{p,q} is created. The problem of bounding W_n(p,q), and its natural hypergraph generalization, was introduced by Balogh, Bollob\\'as, Morris and Riordan. Their main result, specialized to graphs, used algebraic methods to determine W_n(1,q).\n  Our main results in this paper give exact bounds for W_n(p,q), its hypergraph analog"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3598","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.3598","created_at":"2026-05-18T03:01:36.005326+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.3598v2","created_at":"2026-05-18T03:01:36.005326+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3598","created_at":"2026-05-18T03:01:36.005326+00:00"},{"alias_kind":"pith_short_12","alias_value":"GGQPKITEFHNN","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"GGQPKITEFHNNYZOY","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"GGQPKITE","created_at":"2026-05-18T12:27:06.952714+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA","json":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA.json","graph_json":"https://pith.science/api/pith-number/GGQPKITEFHNNYZOYUDHMIY6POA/graph.json","events_json":"https://pith.science/api/pith-number/GGQPKITEFHNNYZOYUDHMIY6POA/events.json","paper":"https://pith.science/paper/GGQPKITE"},"agent_actions":{"view_html":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA","download_json":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA.json","view_paper":"https://pith.science/paper/GGQPKITE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.3598&json=true","fetch_graph":"https://pith.science/api/pith-number/GGQPKITEFHNNYZOYUDHMIY6POA/graph.json","fetch_events":"https://pith.science/api/pith-number/GGQPKITEFHNNYZOYUDHMIY6POA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA/action/storage_attestation","attest_author":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA/action/author_attestation","sign_citation":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA/action/citation_signature","submit_replication":"https://pith.science/pith/GGQPKITEFHNNYZOYUDHMIY6POA/action/replication_record"}},"created_at":"2026-05-18T03:01:36.005326+00:00","updated_at":"2026-05-18T03:01:36.005326+00:00"}