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Let H_{k-1}(Y;R) denote the (k-1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. Let R and k \\geq 1 be fixed. It is shown that p=(k \\log n)/n is a sharp threshold for the vanishing of H_{k-1}(Y;R)."},"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":"math/0609773","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2006-09-27T21:05:16Z","cross_cats_sorted":[],"title_canon_sha256":"aa1ff803056a81975f83ac766656c05fb9d9a1e053b9a211958dd7adf01780aa","abstract_canon_sha256":"8e3295722604b357e3ab9c99f981c0d6b10206e5115c15a4d8dcf072a9a8e482"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:56:36.858058Z","signature_b64":"3j613VXPiIqL403fXv28YcfKZZbpciPNVv6iyF212yimR2fxP7wx2Ae5FafJQLquCbcIQtrujieUkYWmJpshCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2304434ebb46acd601c2d6008eb37ebec76c0a047194ee8757783a29a3ae6468","last_reissued_at":"2026-07-04T14:56:36.857629Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:56:36.857629Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homological connectivity of random k-dimensional complexes","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"N. Wallach, R. Meshulam","submitted_at":"2006-09-27T21:05:16Z","abstract_excerpt":"Let Delta_{n-1} denote the (n-1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Delta_{n-1} obtained by starting with the full (k-1)-dimensional skeleton of Delta_{n-1} and then adding each k-simplex independently with probability p. Let H_{k-1}(Y;R) denote the (k-1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. Let R and k \\geq 1 be fixed. It is shown that p=(k \\log n)/n is a sharp threshold for the vanishing of H_{k-1}(Y;R)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609773","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0609773/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"math/0609773","created_at":"2026-07-04T14:56:36.857701+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0609773v1","created_at":"2026-07-04T14:56:36.857701+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609773","created_at":"2026-07-04T14:56:36.857701+00:00"},{"alias_kind":"pith_short_12","alias_value":"EMCEGTV3I2WN","created_at":"2026-07-04T14:56:36.857701+00:00"},{"alias_kind":"pith_short_16","alias_value":"EMCEGTV3I2WNMAOC","created_at":"2026-07-04T14:56:36.857701+00:00"},{"alias_kind":"pith_short_8","alias_value":"EMCEGTV3","created_at":"2026-07-04T14:56:36.857701+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.15056","citing_title":"Transversal non-Clifford gates on qLDPC codes breaking the $\\sqrt{N}$ distance barrier and quantum-inspired geometry with $\\mathbb{Z}_2$ systolic freedom","ref_index":42,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3","json":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3.json","graph_json":"https://pith.science/api/pith-number/EMCEGTV3I2WNMAOC2YAI5M36X3/graph.json","events_json":"https://pith.science/api/pith-number/EMCEGTV3I2WNMAOC2YAI5M36X3/events.json","paper":"https://pith.science/paper/EMCEGTV3"},"agent_actions":{"view_html":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3","download_json":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3.json","view_paper":"https://pith.science/paper/EMCEGTV3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0609773&json=true","fetch_graph":"https://pith.science/api/pith-number/EMCEGTV3I2WNMAOC2YAI5M36X3/graph.json","fetch_events":"https://pith.science/api/pith-number/EMCEGTV3I2WNMAOC2YAI5M36X3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3/action/storage_attestation","attest_author":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3/action/author_attestation","sign_citation":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3/action/citation_signature","submit_replication":"https://pith.science/pith/EMCEGTV3I2WNMAOC2YAI5M36X3/action/replication_record"}},"created_at":"2026-07-04T14:56:36.857701+00:00","updated_at":"2026-07-04T14:56:36.857701+00:00"}