{"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. For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2.","one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"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.","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"}