{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GCFO5WZBRMO7KTYVNIJW3Q5QTC","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":"40ed81469b714565c0505a2797ef2e9ba3b93eb2853a57bd79bf6c3416682824","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-02T16:06:37Z","title_canon_sha256":"9efe221f8628e97ee60a84cd7d6a9badc05980e19ecba00f2f4fb7ffa75b8e4c"},"schema_version":"1.0","source":{"id":"1204.0449","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0449","created_at":"2026-05-18T03:54:38Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0449v2","created_at":"2026-05-18T03:54:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0449","created_at":"2026-05-18T03:54:38Z"},{"alias_kind":"pith_short_12","alias_value":"GCFO5WZBRMO7","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GCFO5WZBRMO7KTYV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GCFO5WZB","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:196ce3930c57cb48473d65be639477b8ba5650abae9e30f4b728191c85f24cf9","target":"graph","created_at":"2026-05-18T03:54:38Z","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":"Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability of graph properties. In this paper we propose a unified approach to hyperfiniteness. We establish some new results and give new proofs of theorems of Schramm, Lov\\'asz, Newman-Sohler and Ornstein-Weiss.","authors_text":"Gabor Elek","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-02T16:06:37Z","title":"Finite graphs and amenability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0449","kind":"arxiv","version":2},"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:b0845821ea297333e7eaa681de6c521106b5866230e3d61ca601b817e2abafa9","target":"record","created_at":"2026-05-18T03:54:38Z","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":"40ed81469b714565c0505a2797ef2e9ba3b93eb2853a57bd79bf6c3416682824","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-02T16:06:37Z","title_canon_sha256":"9efe221f8628e97ee60a84cd7d6a9badc05980e19ecba00f2f4fb7ffa75b8e4c"},"schema_version":"1.0","source":{"id":"1204.0449","kind":"arxiv","version":2}},"canonical_sha256":"308aeedb218b1df54f156a136dc3b098a6c766983c0ea105e75aa0616fe4f6fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"308aeedb218b1df54f156a136dc3b098a6c766983c0ea105e75aa0616fe4f6fc","first_computed_at":"2026-05-18T03:54:38.960216Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:38.960216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZkVvG965uc4EPIhTjgmJc/zpChEH36JdV7iLZaqQZmyZnVAxJjX57NiiPEOrdI0tuUxbM05htjmMil6oMTKgDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:38.960748Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.0449","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0845821ea297333e7eaa681de6c521106b5866230e3d61ca601b817e2abafa9","sha256:196ce3930c57cb48473d65be639477b8ba5650abae9e30f4b728191c85f24cf9"],"state_sha256":"d1535978a072e457e5040f548ff5d660de5e0e0a80fa1dd418f70c4103d4cffc"}