{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:5ZHDGCFNDEXM2QWS6TFFBF4YAA","short_pith_number":"pith:5ZHDGCFN","canonical_record":{"source":{"id":"2006.05412","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-09T17:21:26Z","cross_cats_sorted":[],"title_canon_sha256":"e717015561fca7cf7565453c2341c023eda3596226cb3331b845b90c29733454","abstract_canon_sha256":"d480d7b4398cce41c34bbdf849a16957aeaec56091923a1458e7469135cce8d5"},"schema_version":"1.0"},"canonical_sha256":"ee4e3308ad192ecd42d2f4ca50979800104bf3bac10d458bb368364172e2b830","source":{"kind":"arxiv","id":"2006.05412","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.05412","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"arxiv_version","alias_value":"2006.05412v2","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.05412","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_12","alias_value":"5ZHDGCFNDEXM","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_16","alias_value":"5ZHDGCFNDEXM2QWS","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_8","alias_value":"5ZHDGCFN","created_at":"2026-07-05T02:56:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:5ZHDGCFNDEXM2QWS6TFFBF4YAA","target":"record","payload":{"canonical_record":{"source":{"id":"2006.05412","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-09T17:21:26Z","cross_cats_sorted":[],"title_canon_sha256":"e717015561fca7cf7565453c2341c023eda3596226cb3331b845b90c29733454","abstract_canon_sha256":"d480d7b4398cce41c34bbdf849a16957aeaec56091923a1458e7469135cce8d5"},"schema_version":"1.0"},"canonical_sha256":"ee4e3308ad192ecd42d2f4ca50979800104bf3bac10d458bb368364172e2b830","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:56:53.131013Z","signature_b64":"vH7zCNv6I2JYc60any0Zo1eyxGnLPdG83GlnfMfR35bidUKN3U1HvV2Cby/Yqu2AewzoZiJ8y9jNXWfUzdkyBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee4e3308ad192ecd42d2f4ca50979800104bf3bac10d458bb368364172e2b830","last_reissued_at":"2026-07-05T02:56:53.130591Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:56:53.130591Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.05412","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":"2026-07-05T02:56:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ycEgb+q/4dFStJSHZQ8knQyStRhVwpnKvz3BngKCNqG3jXIY9UVq0GByjWgzNyoIZk/R8qg84Ww4pHcc5gIEAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T06:56:29.534934Z"},"content_sha256":"8a70f4abfa3122a59961e9ef23adbd858bca37fa393868550120b44d1f5f87f5","schema_version":"1.0","event_id":"sha256:8a70f4abfa3122a59961e9ef23adbd858bca37fa393868550120b44d1f5f87f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:5ZHDGCFNDEXM2QWS6TFFBF4YAA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random Van der Waerden Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ohad Zohar","submitted_at":"2020-06-09T17:21:26Z","abstract_excerpt":"In this paper we prove the Random Van der Waerden Theorem: For $q_1 \\geq q_2 \\geq \\dotsb \\geq q_r \\geq 3 \\in \\mathbb{N}$ there exist $c,C >0$ such that \\[ \\lim_{n \\to \\infty} \\mathbb{P}([n]_p \\rightarrow (q_1,\\dotsc, q_r)) = \\begin{cases} 1 & \\text{if } p \\geq C \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, 0 & \\text{if } p \\leq c \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, \\end{cases}\\] extending the results of R\\\"odl and Ruci\\'nski for the symmetric case $q_i = q$. The proof for the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof for the 0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.05412","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.05412/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:56:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NhkLIZsErvUKhr/LtT6oQKAa6HYcdvXAgcuBJD/m67rCFPA+hayRZ4wRR6pKIMTpL7OSOlbqI7Ud05Jml6HtCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T06:56:29.535293Z"},"content_sha256":"86f425819906bc604418878285ffacf3736aef3dc20cd089605d6ad06191ae66","schema_version":"1.0","event_id":"sha256:86f425819906bc604418878285ffacf3736aef3dc20cd089605d6ad06191ae66"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/bundle.json","state_url":"https://pith.science/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/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":"2026-07-05T06:56:29Z","links":{"resolver":"https://pith.science/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA","bundle":"https://pith.science/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/bundle.json","state":"https://pith.science/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5ZHDGCFNDEXM2QWS6TFFBF4YAA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:5ZHDGCFNDEXM2QWS6TFFBF4YAA","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":"d480d7b4398cce41c34bbdf849a16957aeaec56091923a1458e7469135cce8d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-09T17:21:26Z","title_canon_sha256":"e717015561fca7cf7565453c2341c023eda3596226cb3331b845b90c29733454"},"schema_version":"1.0","source":{"id":"2006.05412","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.05412","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"arxiv_version","alias_value":"2006.05412v2","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.05412","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_12","alias_value":"5ZHDGCFNDEXM","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_16","alias_value":"5ZHDGCFNDEXM2QWS","created_at":"2026-07-05T02:56:53Z"},{"alias_kind":"pith_short_8","alias_value":"5ZHDGCFN","created_at":"2026-07-05T02:56:53Z"}],"graph_snapshots":[{"event_id":"sha256:86f425819906bc604418878285ffacf3736aef3dc20cd089605d6ad06191ae66","target":"graph","created_at":"2026-07-05T02:56:53Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.05412/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we prove the Random Van der Waerden Theorem: For $q_1 \\geq q_2 \\geq \\dotsb \\geq q_r \\geq 3 \\in \\mathbb{N}$ there exist $c,C >0$ such that \\[ \\lim_{n \\to \\infty} \\mathbb{P}([n]_p \\rightarrow (q_1,\\dotsc, q_r)) = \\begin{cases} 1 & \\text{if } p \\geq C \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, 0 & \\text{if } p \\leq c \\cdot n^{-\\frac{q_2}{q_1(q_2-1)}}, \\end{cases}\\] extending the results of R\\\"odl and Ruci\\'nski for the symmetric case $q_i = q$. The proof for the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof for the 0","authors_text":"Ohad Zohar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-09T17:21:26Z","title":"Random Van der Waerden Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.05412","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":"sha256:8a70f4abfa3122a59961e9ef23adbd858bca37fa393868550120b44d1f5f87f5","target":"record","created_at":"2026-07-05T02:56:53Z","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":"d480d7b4398cce41c34bbdf849a16957aeaec56091923a1458e7469135cce8d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-09T17:21:26Z","title_canon_sha256":"e717015561fca7cf7565453c2341c023eda3596226cb3331b845b90c29733454"},"schema_version":"1.0","source":{"id":"2006.05412","kind":"arxiv","version":2}},"canonical_sha256":"ee4e3308ad192ecd42d2f4ca50979800104bf3bac10d458bb368364172e2b830","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee4e3308ad192ecd42d2f4ca50979800104bf3bac10d458bb368364172e2b830","first_computed_at":"2026-07-05T02:56:53.130591Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:56:53.130591Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vH7zCNv6I2JYc60any0Zo1eyxGnLPdG83GlnfMfR35bidUKN3U1HvV2Cby/Yqu2AewzoZiJ8y9jNXWfUzdkyBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:56:53.131013Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.05412","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a70f4abfa3122a59961e9ef23adbd858bca37fa393868550120b44d1f5f87f5","sha256:86f425819906bc604418878285ffacf3736aef3dc20cd089605d6ad06191ae66"],"state_sha256":"0efc9ef28d100fd4e2d84ae4eaea6315aff651ca53ac3c44bccfe907ae8d0c13"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dBLCgm/+7ZXC4011NV5Xr+PFcgyAty7bSVS79qM5JKz5XcIbMlAh82HJxjCs3yDWZaS1AQgHsdbfJ9HxlvblAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T06:56:29.537422Z","bundle_sha256":"4645e190bbe60138fb805ebbac01a6700fb45513d690e0897abd2f15e3a62906"}}