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It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\\cdot 2^k+1$ for all $k\\ge1$. This new result provides partial evidence fo"},"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":"1709.06130","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-18T19:16:34Z","cross_cats_sorted":[],"title_canon_sha256":"78ee974da41f05257ab56f003e44d87930d9cacf1e8524cadc9d1feb11fe20cd","abstract_canon_sha256":"f6234e28992b128c00e8645efa30fbcd843228a799c6783aa943d34d7b307d37"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:58.399756Z","signature_b64":"V+ib+qGoqKcPQW7DCkap9/fPXAJ/91beNs0LUv2Rs0rUOa9XWmUF6kmKADfBkXNoyxRpLO5gtYGHmO4++FJqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93066b8b486d6d4f8355f7bcc059572ff4b0ba289effd584d2314b76585d1785","last_reissued_at":"2026-05-18T00:33:58.399154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:58.399154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gallai-Ramsey numbers of $C_9$ with multiple colors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Bosse, Zi-Xia Song","submitted_at":"2017-09-18T19:16:34Z","abstract_excerpt":"We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\\cdot 2^k+1$ for all $k\\ge1$. 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