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For q=3, R(C_3k,Z_3)=3k+2 for k≥2 and R(W_3k,Z_3)=3k+1 for k≥2."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."}],"snapshot_sha256":"6059f9b986b475e9cb2c5cb98940c8b0c781ff098592495219e0f68bd9738db5"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Cheng Chi, Jialin He","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","title":"On zero-sum Ramsey numbers of cycles and wheels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14954","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T03:20:31.295043Z","id":"460bad04-3323-4951-9a65-82fb62a3cf08","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"","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.","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."}},"verdict_id":"460bad04-3323-4951-9a65-82fb62a3cf08"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2","target":"record","created_at":"2026-05-17T23:38:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"434b1584765c23e10b729ac46d391bda8adb39d32bf7ef49c4883b55be99946a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T15:25:02Z","title_canon_sha256":"c503f365e8e9fd6a1eb0080731ad670406367cbda2e1fe7c132d7e1698eb76d1"},"schema_version":"1.0","source":{"id":"2605.14954","kind":"arxiv","version":1}},"canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ce65e676070411d0373a06b9bc14eeb02233ced53f67f44413f14c52f7a423d","first_computed_at":"2026-05-17T23:38:55.356961Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:55.356961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r4pg8XgggvaDbBoxJbKo3F2yKkt4a535aZldJvxyQz55DRf7jm4rX6rX6DMpaeXM6dCqWhTcXf7BsUKhlqDHDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:55.357567Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14954","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80f0fb6c27f56efed0d4c1440500a2596204ca05ddbbd7859061be9de4d2fcf2","sha256:ccd8ecd4dbf8af7d88b0038f95cf01c0318659d6f8ee50171d428f6a9164cb07"],"state_sha256":"ccdb03f3113465f0808b8e3f7e74107ae9ac54ed47b52ba4f6e782399de6cc16"}