Claims

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1973_02_01_bondy_erdos: Theorem 4 of Bondy and Erdős (1973): R(C_k,K_n) = (k-1)(n-1)+1 whenever k >= n^2-2, the first infinite range of the identity; refereed and credited by the site's commentary.

2004_04_27_nikiforov: Theorem 1 of Nikiforov's preprint arXiv:math/0404501v1 (journal version Combin. Probab. Comput. 14 (2005)): R(C_k,K_n) = (k-1)(n-1)+1 whenever n >= 4 and k >= 4n+2, a linear threshold; refereed.

2018_07_17_keevash_long_skokan: Theorem 1.1 of Keevash, Long and Skokan (IMRN 2021) gives an absolute C with R(C_k,K_n) = (k-1)(n-1)+1 whenever k is at least C log n / log log n, so the identity holds for all large n; finitely many pairs remain unchecked.

2026_09_25_openai: The OpenAI release's preprint of 25 September 2026 proves R(C_k,K_n) = (k-1)(n-1)+1 for every k at least n at least 3, with R(C_3,K_3) = 6, by a reduction to 3,099 finite pattern instances; accepted on its built Lean proof.

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