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Denote by $w_{\\mathrm{\\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\\chi:[1,w_{\\mathrm{\\mathfrak{z}}}(k;r)] \\rightarrow \\{0,1,\\dots,r-1\\}$ admits a $k$-term arithmetic progression $a,a+d,\\dots,a+(k-1)d$ with $\\sum_{j=0}^{k-1} \\chi(a+jd) \\equiv 0 \\,(\\mathrm{mod }\\,r)$. We investigate these numbers as well as a \"mixed\" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\\mathrm{\\mathfrak{z}}}(k;r)$."},"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":"1802.03387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-09T18:50:03Z","cross_cats_sorted":[],"title_canon_sha256":"e4e47f78bf04b16526f144f65d1e58c970d0ee9f0eca0b8e60c3d76de02eb733","abstract_canon_sha256":"4f107b9382513f30ad05e1fe0cb4b2d329a1463191a8679e7bb4701a11b0954a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:57.759310Z","signature_b64":"3j+H5kSRru1teBO3P2aW5QrzKYK0zBbOOTR5AJCsFqjhxNvxPRfjK0sfUXgF5eBzHyZa/cNmjsaw5LTWv+dOBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a66d3b556cd2526dff14642f9ac58cc979086ce8942d09300e8078afe9af6512","last_reissued_at":"2026-05-18T00:23:57.758823Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:57.758823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-sum Analogues of van der Waerden's Theorem on Arithmetic Progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Robertson","submitted_at":"2018-02-09T18:50:03Z","abstract_excerpt":"Let $r$ and $k$ be positive integers with $r \\mid k$. Denote by $w_{\\mathrm{\\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\\chi:[1,w_{\\mathrm{\\mathfrak{z}}}(k;r)] \\rightarrow \\{0,1,\\dots,r-1\\}$ admits a $k$-term arithmetic progression $a,a+d,\\dots,a+(k-1)d$ with $\\sum_{j=0}^{k-1} \\chi(a+jd) \\equiv 0 \\,(\\mathrm{mod }\\,r)$. We investigate these numbers as well as a \"mixed\" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\\mathrm{\\mathfrak{z}}}(k;r)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03387","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":"1802.03387","created_at":"2026-05-18T00:23:57.758885+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.03387v1","created_at":"2026-05-18T00:23:57.758885+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.03387","created_at":"2026-05-18T00:23:57.758885+00:00"},{"alias_kind":"pith_short_12","alias_value":"UZWTWVLM2JJG","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"UZWTWVLM2JJG37YU","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"UZWTWVLM","created_at":"2026-05-18T12:32:56.356000+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/UZWTWVLM2JJG37YUMQXZVRMMZF","json":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF.json","graph_json":"https://pith.science/api/pith-number/UZWTWVLM2JJG37YUMQXZVRMMZF/graph.json","events_json":"https://pith.science/api/pith-number/UZWTWVLM2JJG37YUMQXZVRMMZF/events.json","paper":"https://pith.science/paper/UZWTWVLM"},"agent_actions":{"view_html":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF","download_json":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF.json","view_paper":"https://pith.science/paper/UZWTWVLM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.03387&json=true","fetch_graph":"https://pith.science/api/pith-number/UZWTWVLM2JJG37YUMQXZVRMMZF/graph.json","fetch_events":"https://pith.science/api/pith-number/UZWTWVLM2JJG37YUMQXZVRMMZF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF/action/storage_attestation","attest_author":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF/action/author_attestation","sign_citation":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF/action/citation_signature","submit_replication":"https://pith.science/pith/UZWTWVLM2JJG37YUMQXZVRMMZF/action/replication_record"}},"created_at":"2026-05-18T00:23:57.758885+00:00","updated_at":"2026-05-18T00:23:57.758885+00:00"}