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Define c_2(n,F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. This is a natural problem intermediate (but distinct) from the well-studied Tur\\'an problems and tiling problems.\n  In this paper, we determine c_2(n, K_4) (for n>98) and the associated extremal configurations (for n>998), where K_4 denotes "},"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":"1512.01144","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T16:26:14Z","cross_cats_sorted":[],"title_canon_sha256":"07aa342a2db57a43d2d19e33c1e914c6cb082c3b9be0ee9903452cf7821a57ba","abstract_canon_sha256":"50347b251000edf8142fc1c5ab2ae4eed9994886eef97f9745db22d560d2703f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:18.834142Z","signature_b64":"nCVdeHE2nXhhDsSwVZXqY2BiKZ7eoAm4Y0jptTRgk2gq63J7FbR4z7EiDT7Ej8fd0tyW6Ud61FZCXcDFV1kxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"791e1bb7e74b6683b098d7dfd554fbc0180fd612204b584a9a2ae2bdda9e7279","last_reissued_at":"2026-05-18T01:25:18.833267Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:18.833267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Codegree thresholds for covering 3-uniform hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Victor Falgas-Ravry, Yi Zhao","submitted_at":"2015-12-03T16:26:14Z","abstract_excerpt":"Given two 3-uniform hypergraphs F and G, we say that G has an F-covering if we can cover V(G) by copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V(G) is contained in at least d triples from E(G). Define c_2(n,F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. This is a natural problem intermediate (but distinct) from the well-studied Tur\\'an problems and tiling problems.\n  In this paper, we determine c_2(n, K_4) (for n>98) and the associated extremal configurations (for n>998), where K_4 denotes "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01144","kind":"arxiv","version":1},"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":"1512.01144","created_at":"2026-05-18T01:25:18.833442+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.01144v1","created_at":"2026-05-18T01:25:18.833442+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01144","created_at":"2026-05-18T01:25:18.833442+00:00"},{"alias_kind":"pith_short_12","alias_value":"PEPBXN7HJNTI","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"PEPBXN7HJNTIHMEY","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"PEPBXN7H","created_at":"2026-05-18T12:29:37.295048+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/PEPBXN7HJNTIHMEY27P5KVH3YA","json":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA.json","graph_json":"https://pith.science/api/pith-number/PEPBXN7HJNTIHMEY27P5KVH3YA/graph.json","events_json":"https://pith.science/api/pith-number/PEPBXN7HJNTIHMEY27P5KVH3YA/events.json","paper":"https://pith.science/paper/PEPBXN7H"},"agent_actions":{"view_html":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA","download_json":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA.json","view_paper":"https://pith.science/paper/PEPBXN7H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.01144&json=true","fetch_graph":"https://pith.science/api/pith-number/PEPBXN7HJNTIHMEY27P5KVH3YA/graph.json","fetch_events":"https://pith.science/api/pith-number/PEPBXN7HJNTIHMEY27P5KVH3YA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA/action/storage_attestation","attest_author":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA/action/author_attestation","sign_citation":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA/action/citation_signature","submit_replication":"https://pith.science/pith/PEPBXN7HJNTIHMEY27P5KVH3YA/action/replication_record"}},"created_at":"2026-05-18T01:25:18.833442+00:00","updated_at":"2026-05-18T01:25:18.833442+00:00"}