{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:5MQHZVKRZVRSLA5D6M77JHD57A","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":"0279662e8ec25065e0d54820f901bb6cb9d52534107d2f4016860028069af132","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-09-13T12:59:28Z","title_canon_sha256":"a97b71f5c0205369f1f5b9f14a4a5641a5fc2698f3c3334f3ec172668d1d933a"},"schema_version":"1.0","source":{"id":"0809.2335","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.2335","created_at":"2026-05-18T04:25:39Z"},{"alias_kind":"arxiv_version","alias_value":"0809.2335v5","created_at":"2026-05-18T04:25:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.2335","created_at":"2026-05-18T04:25:39Z"},{"alias_kind":"pith_short_12","alias_value":"5MQHZVKRZVRS","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"5MQHZVKRZVRSLA5D","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"5MQHZVKR","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:f6a0e2135798559ea3e9360082f2e9da3f91cd998f2839712d3de2d43158fed6","target":"graph","created_at":"2026-05-18T04:25:39Z","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":"We study some percolation problems on the complete graph over $\\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.","authors_text":"A. Berarducci, M. Novaga, P. Majer","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-09-13T12:59:28Z","title":"Infinite paths and cliques in random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.2335","kind":"arxiv","version":5},"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:e7c25ff3302212e20a1b1717f231b76b00d12a7f995c59a71ec4f1cd4c869eb9","target":"record","created_at":"2026-05-18T04:25:39Z","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":"0279662e8ec25065e0d54820f901bb6cb9d52534107d2f4016860028069af132","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-09-13T12:59:28Z","title_canon_sha256":"a97b71f5c0205369f1f5b9f14a4a5641a5fc2698f3c3334f3ec172668d1d933a"},"schema_version":"1.0","source":{"id":"0809.2335","kind":"arxiv","version":5}},"canonical_sha256":"eb207cd551cd632583a3f33ff49c7df81e81b2f7d0eaa24c57290015a0cb8b59","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb207cd551cd632583a3f33ff49c7df81e81b2f7d0eaa24c57290015a0cb8b59","first_computed_at":"2026-05-18T04:25:39.343111Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:39.343111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RS8Zw1PW2bMWMJswNICgBnosNK98r+HndKabsWZM/fTD8FFq8vDMlWqM2tK0KL9inj5QZUqLS3UiQl4aH+azDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:39.343702Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.2335","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e7c25ff3302212e20a1b1717f231b76b00d12a7f995c59a71ec4f1cd4c869eb9","sha256:f6a0e2135798559ea3e9360082f2e9da3f91cd998f2839712d3de2d43158fed6"],"state_sha256":"158c5584fa58f76950e80eff8a65a3512fca15d4e74666745dc798d3f1840fe8"}