{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JVS53GHT6Q43Q62G5GKYXQ63F6","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":"24a4bd24663ffc424c60c54c7780e4e92fd26a2005b600038f84af5323a03d2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-11T22:57:06Z","title_canon_sha256":"c4186918f7bd1ebec13b8834ed341fa87382234e8eb20c6dd4cb11c19a28f1fd"},"schema_version":"1.0","source":{"id":"1203.2383","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.2383","created_at":"2026-05-18T04:00:25Z"},{"alias_kind":"arxiv_version","alias_value":"1203.2383v1","created_at":"2026-05-18T04:00:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.2383","created_at":"2026-05-18T04:00:25Z"},{"alias_kind":"pith_short_12","alias_value":"JVS53GHT6Q43","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JVS53GHT6Q43Q62G","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JVS53GHT","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:24adc5c674b1d2ceb49f4d9e49ad2e5a5ac20fe4d87401531a53e24fdd5d4052","target":"graph","created_at":"2026-05-18T04:00:25Z","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 $G$ be a finite abelian group with exponent $n$, and let $r$ be a positive integer. Let $A$ be a $k\\times m$ matrix with integer entries. We show that if $A$ satisfies some natural conditions and $|G|$ is large enough then, for each $r$--coloring of $G\\setminus \\{0\\}$, there is $\\delta$ depending only on $r,n$ and $m$ such that the homogeneous linear system $Ax=0$ has at least $\\delta |G|^{m-k}$ monochromatic solutions. Density versions of this counting result are also addressed.","authors_text":"Llu\\'is Vena, Oriol Serra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-11T22:57:06Z","title":"On the number of monochromatic solutions of integer linear systems on Abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.2383","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":"sha256:8dfb3e1e194b43319ed2711ac9c695fd149536da2277a96767a91918af6a3292","target":"record","created_at":"2026-05-18T04:00:25Z","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":"24a4bd24663ffc424c60c54c7780e4e92fd26a2005b600038f84af5323a03d2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-11T22:57:06Z","title_canon_sha256":"c4186918f7bd1ebec13b8834ed341fa87382234e8eb20c6dd4cb11c19a28f1fd"},"schema_version":"1.0","source":{"id":"1203.2383","kind":"arxiv","version":1}},"canonical_sha256":"4d65dd98f3f439b87b46e9958bc3db2fbb8c609238279a10211f6a27b7c8ac54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d65dd98f3f439b87b46e9958bc3db2fbb8c609238279a10211f6a27b7c8ac54","first_computed_at":"2026-05-18T04:00:25.648234Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:25.648234Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QOyM5HijLZq3cSEIybsxHKbMx2jtk77P2yKkMLdqpJh3rFu/9R4ys2rOW9U5EEBqXMUCD4EZWoTaEfatJ8jrBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:25.648975Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.2383","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8dfb3e1e194b43319ed2711ac9c695fd149536da2277a96767a91918af6a3292","sha256:24adc5c674b1d2ceb49f4d9e49ad2e5a5ac20fe4d87401531a53e24fdd5d4052"],"state_sha256":"5a2e02a769109b5a1a81ed5c68847067240ac18923d8927bd88a1a73de1f3068"}