{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:LNMZHD3RQOOALJD2H7MBKJRMJE","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":"ffaa0c8d8aecade2f294065f97472d3504d725d2a1fb4e6115acbc0a71ef3363","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-24T14:06:38Z","title_canon_sha256":"863a5a2808a9790ae5b20b945fcfa955fca4cd1f6eb09858c14c62ff68ec6829"},"schema_version":"1.0","source":{"id":"1402.5829","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5829","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5829v2","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5829","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"LNMZHD3RQOOA","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"LNMZHD3RQOOALJD2","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"LNMZHD3R","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:a32bd4869fb5ce977e2183c9f943a6fff7a040c2634dd472855e987ec7dbb064","target":"graph","created_at":"2026-05-18T02:56:44Z","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":"In 1982, Beutelspacher and Brestovansky determined the 2-color Rado number of the equation $$x_1+x_2+\\cdots +x_{m-1}=x_m$$ for all $m\\geq 3.$ Here we extend their result by determining the 2-color Rado number of the equation $$x_1+x_2+\\cdots +x_n=y_1+y_2+\\cdots +y_k$$ for all $n\\geq 2$ and $k\\geq 2.$ As a consequence, we determine the 2-color Rado number of $$x_1+x_2+\\cdots +x_n=a_1y_1+\\cdots +a_{\\ell}y_{\\ell}$$ in all cases where $n\\geq 2$ and $n\\geq a_1+\\cdots +a_{\\ell},$ and in most cases where $n\\geq 2$ and $2n\\geq a_1+\\cdots +a_{\\ell}.$","authors_text":"Dan Saracino","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-24T14:06:38Z","title":"The 2-color Rado number of $x_1+x_2+\\cdots +x_n=y_1+y_2+\\cdots +y_k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5829","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:b1518cf5d0ed04642cad98d5f57e446ad96201d81ff8d036ba238c5086af5e1d","target":"record","created_at":"2026-05-18T02:56:44Z","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":"ffaa0c8d8aecade2f294065f97472d3504d725d2a1fb4e6115acbc0a71ef3363","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-24T14:06:38Z","title_canon_sha256":"863a5a2808a9790ae5b20b945fcfa955fca4cd1f6eb09858c14c62ff68ec6829"},"schema_version":"1.0","source":{"id":"1402.5829","kind":"arxiv","version":2}},"canonical_sha256":"5b59938f71839c05a47a3fd815262c490495363f546c8818a13d62d1ccc6c8b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b59938f71839c05a47a3fd815262c490495363f546c8818a13d62d1ccc6c8b8","first_computed_at":"2026-05-18T02:56:44.998530Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:44.998530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CcqRHatPpNQcm9FpWA2fMnvxpDCldjhBOM8lz/qfknlIpx8pDkzUTEOGPiv1qscc1oMIU3LbCE4Tw5/EOIwvAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:44.999463Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.5829","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1518cf5d0ed04642cad98d5f57e446ad96201d81ff8d036ba238c5086af5e1d","sha256:a32bd4869fb5ce977e2183c9f943a6fff7a040c2634dd472855e987ec7dbb064"],"state_sha256":"eeb98b6bb4f7f354157d3ec3debc43fafcb52b38fce832b79115b14e356dbfc5"}