{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:7XMEWE27RYQQY4ZDXMBWNEHRBB","short_pith_number":"pith:7XMEWE27","canonical_record":{"source":{"id":"0707.3888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-07-26T10:03:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"846c3b8893dbee01ccab3ff56d00cf226184077e7b2b94b18762c914dfe12dd6","abstract_canon_sha256":"34c6f313538f1525df7542c7acf3a2718be223a33aa56c094e357cea6538b30c"},"schema_version":"1.0"},"canonical_sha256":"fdd84b135f8e210c7323bb036690f1086e29faba417964cb8a3e39b9fd7ea739","source":{"kind":"arxiv","id":"0707.3888","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0707.3888","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"arxiv_version","alias_value":"0707.3888v2","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.3888","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"pith_short_12","alias_value":"7XMEWE27RYQQ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"7XMEWE27RYQQY4ZD","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"7XMEWE27","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:7XMEWE27RYQQY4ZDXMBWNEHRBB","target":"record","payload":{"canonical_record":{"source":{"id":"0707.3888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-07-26T10:03:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"846c3b8893dbee01ccab3ff56d00cf226184077e7b2b94b18762c914dfe12dd6","abstract_canon_sha256":"34c6f313538f1525df7542c7acf3a2718be223a33aa56c094e357cea6538b30c"},"schema_version":"1.0"},"canonical_sha256":"fdd84b135f8e210c7323bb036690f1086e29faba417964cb8a3e39b9fd7ea739","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:22.389182Z","signature_b64":"j6/vivBQjlfUiJ1mKSEsn1jVwjVJ0hDtGUcaJybebOO4foZwZgJUGY3hpXw+CoQ8hcR+k5FNXzDjnkJ+3R7KDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fdd84b135f8e210c7323bb036690f1086e29faba417964cb8a3e39b9fd7ea739","last_reissued_at":"2026-05-18T03:55:22.388458Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:22.388458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0707.3888","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:55:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2SYSDI5+rUnF0pqSrB1b+Pq76ZAPCmqt1rguesAMKfPTfwXVLnl+ux5GbUeczjeczXZn8T0jTzirn5p/pfLAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:17:57.002348Z"},"content_sha256":"8327dd02a2fbf715b353d7e3dca94c473a8bcb0c0d5535972e83794e65207cae","schema_version":"1.0","event_id":"sha256:8327dd02a2fbf715b353d7e3dca94c473a8bcb0c0d5535972e83794e65207cae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:7XMEWE27RYQQY4ZDXMBWNEHRBB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal Arithmetic Progressions in Random Subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ariel Yadin, Itai Benjamini, Ofer Zeitouni","submitted_at":"2007-07-26T10:03:28Z","abstract_excerpt":"Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2))."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.3888","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:55:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SEbnDJ3gUMiycaE0dAuGWS8TxDO4Qw8+t1HELJrOaPGJKe67iCWNUlcAP6T9LjaScw1+Rm1Hi58+ej919FXFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:17:57.003015Z"},"content_sha256":"ca595d9a0684b14bccdcb4e240074409675f9a5f178370a566ab6d4debb2a785","schema_version":"1.0","event_id":"sha256:ca595d9a0684b14bccdcb4e240074409675f9a5f178370a566ab6d4debb2a785"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/bundle.json","state_url":"https://pith.science/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T19:17:57Z","links":{"resolver":"https://pith.science/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB","bundle":"https://pith.science/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/bundle.json","state":"https://pith.science/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7XMEWE27RYQQY4ZDXMBWNEHRBB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:7XMEWE27RYQQY4ZDXMBWNEHRBB","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":"34c6f313538f1525df7542c7acf3a2718be223a33aa56c094e357cea6538b30c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-07-26T10:03:28Z","title_canon_sha256":"846c3b8893dbee01ccab3ff56d00cf226184077e7b2b94b18762c914dfe12dd6"},"schema_version":"1.0","source":{"id":"0707.3888","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0707.3888","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"arxiv_version","alias_value":"0707.3888v2","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.3888","created_at":"2026-05-18T03:55:22Z"},{"alias_kind":"pith_short_12","alias_value":"7XMEWE27RYQQ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"7XMEWE27RYQQY4ZD","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"7XMEWE27","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:ca595d9a0684b14bccdcb4e240074409675f9a5f178370a566ab6d4debb2a785","target":"graph","created_at":"2026-05-18T03:55:22Z","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":"Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2)).","authors_text":"Ariel Yadin, Itai Benjamini, Ofer Zeitouni","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-07-26T10:03:28Z","title":"Maximal Arithmetic Progressions in Random Subsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.3888","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":"sha256:8327dd02a2fbf715b353d7e3dca94c473a8bcb0c0d5535972e83794e65207cae","target":"record","created_at":"2026-05-18T03:55:22Z","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":"34c6f313538f1525df7542c7acf3a2718be223a33aa56c094e357cea6538b30c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-07-26T10:03:28Z","title_canon_sha256":"846c3b8893dbee01ccab3ff56d00cf226184077e7b2b94b18762c914dfe12dd6"},"schema_version":"1.0","source":{"id":"0707.3888","kind":"arxiv","version":2}},"canonical_sha256":"fdd84b135f8e210c7323bb036690f1086e29faba417964cb8a3e39b9fd7ea739","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fdd84b135f8e210c7323bb036690f1086e29faba417964cb8a3e39b9fd7ea739","first_computed_at":"2026-05-18T03:55:22.388458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:22.388458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j6/vivBQjlfUiJ1mKSEsn1jVwjVJ0hDtGUcaJybebOO4foZwZgJUGY3hpXw+CoQ8hcR+k5FNXzDjnkJ+3R7KDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:22.389182Z","signed_message":"canonical_sha256_bytes"},"source_id":"0707.3888","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8327dd02a2fbf715b353d7e3dca94c473a8bcb0c0d5535972e83794e65207cae","sha256:ca595d9a0684b14bccdcb4e240074409675f9a5f178370a566ab6d4debb2a785"],"state_sha256":"6b356673820cbea10ddd4daad1f5b3dbdc35ef7e059957019a21bb105f5391dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"daWkKvjPTj9d1JkgeVLEbpE4WOwZD8CunZ4cD1Fxl1r4Ru7MQx3oyjAtqpRTosA27QV32Mu0Gs9/JrVJA7cABQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T19:17:57.006604Z","bundle_sha256":"56f344db8e399f7f1e9a1ae6959d54761c93f43d08945c609454c5dfbc8930a0"}}