{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:DUJ6SHEZO6SQLZV2VY2Z3X2SYX","short_pith_number":"pith:DUJ6SHEZ","schema_version":"1.0","canonical_sha256":"1d13e91c9977a505e6baae359ddf52c5e1948f179e4dfa74103cf739baff78e7","source":{"kind":"arxiv","id":"1506.04686","version":2},"attestation_state":"computed","paper":{"title":"Extremal Bounds for Bootstrap Percolation in the Hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan A. Noel, Natasha Morrison","submitted_at":"2015-06-15T18:03:53Z","abstract_excerpt":"The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A_0$ of \"infected\" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates.\n  We prove a conjecture of Balogh and Bollob\\'as which says that, for fixed $r$ and $d\\to\\infty$, every percolating set in the $d$-dimensional hypercube has cardinality at least $\\frac{1+o(1)}{r}\\binom{d}{r-1}$. We also "},"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":"1506.04686","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-06-15T18:03:53Z","cross_cats_sorted":[],"title_canon_sha256":"d4795af77fe7b05272835b0e404fbab3bb00d5e5dac5d26e5d5369e53a00c433","abstract_canon_sha256":"d1a481ace23ecd22971b44b8fc670e780ff539018619c1aa4206999d97017876"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:31.835899Z","signature_b64":"zhCcqyulQZ++ii3yQRSexwzoi3icOh5PtkcYuTAp9z3Yczmu8t88UoBPR+X5a68+cKM1wIBZPW5iS3SWeHhUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d13e91c9977a505e6baae359ddf52c5e1948f179e4dfa74103cf739baff78e7","last_reissued_at":"2026-05-18T00:31:31.835400Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:31.835400Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal Bounds for Bootstrap Percolation in the Hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan A. Noel, Natasha Morrison","submitted_at":"2015-06-15T18:03:53Z","abstract_excerpt":"The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A_0$ of \"infected\" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates.\n  We prove a conjecture of Balogh and Bollob\\'as which says that, for fixed $r$ and $d\\to\\infty$, every percolating set in the $d$-dimensional hypercube has cardinality at least $\\frac{1+o(1)}{r}\\binom{d}{r-1}$. We also "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04686","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":"1506.04686","created_at":"2026-05-18T00:31:31.835482+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.04686v2","created_at":"2026-05-18T00:31:31.835482+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04686","created_at":"2026-05-18T00:31:31.835482+00:00"},{"alias_kind":"pith_short_12","alias_value":"DUJ6SHEZO6SQ","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DUJ6SHEZO6SQLZV2","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DUJ6SHEZ","created_at":"2026-05-18T12:29:17.054201+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/DUJ6SHEZO6SQLZV2VY2Z3X2SYX","json":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX.json","graph_json":"https://pith.science/api/pith-number/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/graph.json","events_json":"https://pith.science/api/pith-number/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/events.json","paper":"https://pith.science/paper/DUJ6SHEZ"},"agent_actions":{"view_html":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX","download_json":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX.json","view_paper":"https://pith.science/paper/DUJ6SHEZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.04686&json=true","fetch_graph":"https://pith.science/api/pith-number/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/graph.json","fetch_events":"https://pith.science/api/pith-number/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/action/storage_attestation","attest_author":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/action/author_attestation","sign_citation":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/action/citation_signature","submit_replication":"https://pith.science/pith/DUJ6SHEZO6SQLZV2VY2Z3X2SYX/action/replication_record"}},"created_at":"2026-05-18T00:31:31.835482+00:00","updated_at":"2026-05-18T00:31:31.835482+00:00"}