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We give some formulas and lower bounds for various instances."},"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":"1802.03382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-09T18:46:07Z","cross_cats_sorted":[],"title_canon_sha256":"ef0173aa46836ad9e37f67524e55a49d7a469eef77b069810bfe1177d7735c8e","abstract_canon_sha256":"8e9e33dd207eed8ff0ee9923c834355f3bc0c756d275178d3a62de9ea35b0d1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:57.785525Z","signature_b64":"6MZTvwHBSfuTurVa0z5r3Gux6wok3a19ktv9Xx9SzYN6f3xLcRovJMz+YkPm6VppGnfR5E0Kb3l3MAZnqi+wAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bdec4588568d4d1c7212075caf29a18710a4c5d3c515b84a3b16618f67f3d49","last_reissued_at":"2026-05-18T00:23:57.784930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:57.784930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-sum Generalized Schur Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Robertson","submitted_at":"2018-02-09T18:46:07Z","abstract_excerpt":"Let $r$ and $k$ be positive integers with $r \\mid k$. Denote by $S_{\\mathrm{\\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\\chi:[1,n] \\rightarrow \\{0,1,\\dots,r-1\\}$ admits a solution to $\\sum_{i=1}^{k-1} x_i = x_k$ with $\\sum_{i=1}^{k} \\chi(x_i) \\equiv 0 \\,(\\mathrm{mod }\\,r)$. 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