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For any integer $\\ell \\in [1, k]$ and $\\epsilon \\in \\{0,1\\}$, we let $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}$ be the linear homogeneous equation defined by $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}: x_1 + \\cdots + x_{k-(rv+\\epsilon)} =x_{k-(rv+\\epsilon-1)} +\\cdots+ \\ell x_{k}$. We denote the number $S_{\\mathfrak{z},m}^{(\\ell, \\epsilon)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\\chi: [1, t] \\to \\{0, 1,\\ldo"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.08725","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-27T08:09:44Z","cross_cats_sorted":[],"title_canon_sha256":"7fa29256aee919e69ab362af0c75874411347b16803dad8e2b85d8578266ad1d","abstract_canon_sha256":"5b7989e4c2b8e9de774d6ed4602d9ca522d4865e0418f57e4a90d2e80283481f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:13.458938Z","signature_b64":"gNFeLhilPtPPkm7Sj0AV2Uxyyn/A7+zS+Q57MMeT84Nq77IJh58nnTqOyAduuUaJM1GEnIW0s6Z3EDPYVdzwAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41f4e69feb3ca2718043a52171965f3f72734233d946919c0266fb36d9bce358","last_reissued_at":"2026-05-18T00:07:13.458194Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:13.458194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On determination of Zero-sum $\\ell$-generalized Schur Numbers for some linear equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bidisha Roy, Subha Sarkar","submitted_at":"2018-08-27T08:09:44Z","abstract_excerpt":"Let $r$, $m$ and $k\\geq 2$ be positive integers such that $r\\mid k$ and let $v \\in \\left[ 0,\\lfloor \\frac{k-1}{2r} \\rfloor \\right]$ be any integer. For any integer $\\ell \\in [1, k]$ and $\\epsilon \\in \\{0,1\\}$, we let $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}$ be the linear homogeneous equation defined by $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}: x_1 + \\cdots + x_{k-(rv+\\epsilon)} =x_{k-(rv+\\epsilon-1)} +\\cdots+ \\ell x_{k}$. We denote the number $S_{\\mathfrak{z},m}^{(\\ell, \\epsilon)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\\chi: [1, t] \\to \\{0, 1,\\ldo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08725","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.08725","created_at":"2026-05-18T00:07:13.458300+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.08725v1","created_at":"2026-05-18T00:07:13.458300+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.08725","created_at":"2026-05-18T00:07:13.458300+00:00"},{"alias_kind":"pith_short_12","alias_value":"IH2ONH7LHSRH","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"IH2ONH7LHSRHDACD","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"IH2ONH7L","created_at":"2026-05-18T12:32:28.185984+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5","json":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5.json","graph_json":"https://pith.science/api/pith-number/IH2ONH7LHSRHDACDUUQXDFS7H5/graph.json","events_json":"https://pith.science/api/pith-number/IH2ONH7LHSRHDACDUUQXDFS7H5/events.json","paper":"https://pith.science/paper/IH2ONH7L"},"agent_actions":{"view_html":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5","download_json":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5.json","view_paper":"https://pith.science/paper/IH2ONH7L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.08725&json=true","fetch_graph":"https://pith.science/api/pith-number/IH2ONH7LHSRHDACDUUQXDFS7H5/graph.json","fetch_events":"https://pith.science/api/pith-number/IH2ONH7LHSRHDACDUUQXDFS7H5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5/action/storage_attestation","attest_author":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5/action/author_attestation","sign_citation":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5/action/citation_signature","submit_replication":"https://pith.science/pith/IH2ONH7LHSRHDACDUUQXDFS7H5/action/replication_record"}},"created_at":"2026-05-18T00:07:13.458300+00:00","updated_at":"2026-05-18T00:07:13.458300+00:00"}