{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:R7WO2WQAK3NBGWWMKDI5QFZUBF","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":"54756c4d66ff2c94e93cce28a51fcc757685b1503f08c183d4ae6627e529c640","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-14T18:34:54Z","title_canon_sha256":"ff9bc6fdad5f1497f3fdc08ab39185454beb855c4c1b57fa3a0784e426417bdc"},"schema_version":"1.0","source":{"id":"1107.2888","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2888","created_at":"2026-05-18T04:18:15Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2888v1","created_at":"2026-05-18T04:18:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2888","created_at":"2026-05-18T04:18:15Z"},{"alias_kind":"pith_short_12","alias_value":"R7WO2WQAK3NB","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"R7WO2WQAK3NBGWWM","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"R7WO2WQA","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:f5bac57bb93b67a7a39d202b60eeeb24fbf3a0c66c06e45627fa6bad0dcdf853","target":"graph","created_at":"2026-05-18T04:18:15Z","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":"This paper is motivated by a recent result of Wolf \\cite{wolf} on the minimum number of monochromatic 4-term arithmetic progressions(4-APs, for short) in $\\Z_p$, where $p$ is a prime number. Wolf proved that there is a 2-coloring of $\\Z_p$ with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in $\\Z_n$ for general $n$. We ","authors_text":"Linyuan Lu, Xing Peng","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-14T18:34:54Z","title":"Monochromatic 4-term arithmetic progressions in 2-colorings of $\\mathbb Z_n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2888","kind":"arxiv","version":1},"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:a343b22d617c334c4a7f6c1634835268cc24be03c0543db5bf07b45a2d48f0b0","target":"record","created_at":"2026-05-18T04:18:15Z","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":"54756c4d66ff2c94e93cce28a51fcc757685b1503f08c183d4ae6627e529c640","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-14T18:34:54Z","title_canon_sha256":"ff9bc6fdad5f1497f3fdc08ab39185454beb855c4c1b57fa3a0784e426417bdc"},"schema_version":"1.0","source":{"id":"1107.2888","kind":"arxiv","version":1}},"canonical_sha256":"8feced5a0056da135acc50d1d81734095de0d564e3332d873078c5ff96d219db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8feced5a0056da135acc50d1d81734095de0d564e3332d873078c5ff96d219db","first_computed_at":"2026-05-18T04:18:15.554447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:15.554447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fVHr7J8/KAKxFE/JUwy89GV2fWQoj47MRAxaWkPUdJ0W/P74gxskwF27kgqLX+Vo3Ujp9RypFI8pPhWqzo5lBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:15.554867Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2888","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a343b22d617c334c4a7f6c1634835268cc24be03c0543db5bf07b45a2d48f0b0","sha256:f5bac57bb93b67a7a39d202b60eeeb24fbf3a0c66c06e45627fa6bad0dcdf853"],"state_sha256":"4bd29cfea96148d280635bc36769c4b5a77a65816b3e33f4dd344dc6a5dc4234"}