{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:NTTF4Z3AOBAR2A3TUBVZXQKO5M","short_pith_number":"pith:NTTF4Z3A","schema_version":"1.0","canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","source":{"kind":"arxiv","id":"2605.14954","version":1},"attestation_state":"computed","paper":{"title":"On zero-sum Ramsey numbers of cycles and wheels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Chi, Jialin He","submitted_at":"2026-05-14T15:25:02Z","abstract_excerpt":"For an integer $q\\ge 2$ and a graph $F$ with $q\\mid e(F)$, let $R(F,\\Z_q)$ be the least integer $n$ such that every edge-labeling $w\\colon E(K_n)\\to \\Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\\Z_q)\\le \\max\\{R(C_{2q},\\Z_q),qk+q-1\\}$ via an insertion argument rooted in the classic Erd\\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\\Z_q)\\le \\max\\{35q^2,qk+q-1\\}$ for every $q\\ge 3$. We also show that $R(C_{qk},\\Z_q)\\ge qk+q-1$ for odd $q\\ge 3$. Hence, for every fixed o"},"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":"2605.14954","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1","abstract_canon_sha256":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:55.357567Z","signature_b64":"r4pg8XgggvaDbBoxJbKo3F2yKkt4a535aZldJvxyQz55DRf7jm4rX6rX6DMpaeXM6dCqWhTcXf7BsUKhlqDHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","last_reissued_at":"2026-05-17T23:38:55.356961Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:55.356961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On zero-sum Ramsey numbers of cycles and wheels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cheng Chi, Jialin He","submitted_at":"2026-05-14T15:25:02Z","abstract_excerpt":"For an integer $q\\ge 2$ and a graph $F$ with $q\\mid e(F)$, let $R(F,\\Z_q)$ be the least integer $n$ such that every edge-labeling $w\\colon E(K_n)\\to \\Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\\Z_q)\\le \\max\\{R(C_{2q},\\Z_q),qk+q-1\\}$ via an insertion argument rooted in the classic Erd\\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\\Z_q)\\le \\max\\{35q^2,qk+q-1\\}$ for every $q\\ge 3$. We also show that $R(C_{qk},\\Z_q)\\ge qk+q-1$ for odd $q\\ge 3$. Hence, for every fixed o"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The upper bound for general q relies on Pikhurko's external result that R(C_2q,Z_q)≤35q²; if that bound is loose or inapplicable in the insertion step, the claimed exact value for large k would not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"R(C_qk, Z_q) equals qk + q - 1 exactly for odd q ≥ 3 and k ≥ 35q, with matching exact results for q=3 cycles and wheels W_3k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"6059f9b986b475e9cb2c5cb98940c8b0c781ff098592495219e0f68bd9738db5"},"source":{"id":"2605.14954","kind":"arxiv","version":1},"verdict":{"id":"460bad04-3323-4951-9a65-82fb62a3cf08","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:20:31.295043Z","strongest_claim":"For every fixed odd q≥3 and every k≥35q, we obtain the exact value R(C_qk,Z_q)=qk+q-1. 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