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Recently, this conjecture has been verified for $k=3$ if $n$ is large. In this note, we prove that for every integer $k\\geq 4$, $"},"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":"1005.3926","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-21T10:28:52Z","cross_cats_sorted":[],"title_canon_sha256":"13ff71990b7c023d1fcbb3570e68e1cd4beca081b842ea32d26960be2077646a","abstract_canon_sha256":"fcd51296079f3282d26aa2778def4d77b1cee39a80e850c2f1d4721b6f7c344c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:54.622842Z","signature_b64":"7S2+v9/AzI9reN/FWfGjtcX4Ws+FUg13VCntuHQ+NI1lWVNrVXBBQ/s9llnyGaPUebroNV6yJ/n/hYihYB4iDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5ac96947e7b6e2d8b495cbaa9f5fc2a78acd762b1c55889297ceae4631f7cc0","last_reissued_at":"2026-05-18T04:41:54.622341Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:54.622341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Multi-coloured Ramsey Numbers of Cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jozef Skokan, Mikl\\'os Simonovits, Tomasz {\\L}uczak","submitted_at":"2010-05-21T10:28:52Z","abstract_excerpt":"For a graph $L$ and an integer $k\\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$ as a subgraph.\n  Bondy and Erd\\H{o}s conjectured that for an odd cycle $C_n$ on $n$ vertices, $$R_k(C_n) = 2^{k-1}(n-1)+1 \\text{for $n>3$.}$$ They proved the case when $k=2$ and also provided an upper bound $R_k(C_n)\\leq (k+2)!n$. Recently, this conjecture has been verified for $k=3$ if $n$ is large. In this note, we prove that for every integer $k\\geq 4$, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.3926","kind":"arxiv","version":2},"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":"1005.3926","created_at":"2026-05-18T04:41:54.622421+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.3926v2","created_at":"2026-05-18T04:41:54.622421+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.3926","created_at":"2026-05-18T04:41:54.622421+00:00"},{"alias_kind":"pith_short_12","alias_value":"WWWJNFD6PNXC","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"WWWJNFD6PNXC3C2J","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"WWWJNFD6","created_at":"2026-05-18T12:26:17.028572+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/WWWJNFD6PNXC3C2JLS5KT5P4FJ","json":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ.json","graph_json":"https://pith.science/api/pith-number/WWWJNFD6PNXC3C2JLS5KT5P4FJ/graph.json","events_json":"https://pith.science/api/pith-number/WWWJNFD6PNXC3C2JLS5KT5P4FJ/events.json","paper":"https://pith.science/paper/WWWJNFD6"},"agent_actions":{"view_html":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ","download_json":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ.json","view_paper":"https://pith.science/paper/WWWJNFD6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.3926&json=true","fetch_graph":"https://pith.science/api/pith-number/WWWJNFD6PNXC3C2JLS5KT5P4FJ/graph.json","fetch_events":"https://pith.science/api/pith-number/WWWJNFD6PNXC3C2JLS5KT5P4FJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ/action/storage_attestation","attest_author":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ/action/author_attestation","sign_citation":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ/action/citation_signature","submit_replication":"https://pith.science/pith/WWWJNFD6PNXC3C2JLS5KT5P4FJ/action/replication_record"}},"created_at":"2026-05-18T04:41:54.622421+00:00","updated_at":"2026-05-18T04:41:54.622421+00:00"}