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Claim. Keevash, Long and Skokan's Theorem 1.1 (preprint p. 2; Int. Math. Res. Not. IMRN 2021, no. 1, 275--300) states that there is an absolute constant $C\ge1$ such that

$$R(C_k,K_n)=(k-1)(n-1)+1 \qquad\text{whenever } n\ge3 \text{ and } k\ge C\frac{\log n}{\log\log n},$$

with logarithms to base $2$, in the letters of Problem 551 (the paper writes $r(C_\ell,K_n)$). Since $C\log n/\log\log n<n$ for all $n$ beyond a threshold $n_0(C)$, the theorem proves the problem's identity for every $k\ge n$ once $n\ge n_0(C)$, which the site describes as the conjecture for all large $n$ (the paper says: for large cycle length). The paper does not compute $C$ and remarks (p. 16) that a reasonable value, less than 20, should be obtainable with more work. Its Theorem 1.2 shows the threshold is tight up to the constant: $R(C_k,K_n)>n\log n$ for $3\le k\le(1-\varepsilon)\log n/\log\log n$ and $n\ge n_0(\varepsilon)$, so the formula cannot hold for cycle lengths much below the threshold.

Covers. Every pair $k\ge n\ge n_0(C)$, and for smaller $n$ every $k$ at or above the threshold $C\log n/\log\log n$. The remaining finite check is the set of pairs with $n<n_0(C)$ and $n\le k<C\log n/\log\log n$; the refereed ranges $k\ge n^2-2$ of Bondy and Erdős (the accepted partial claim Bondy and Erdős 1973) and $k\ge4n+2$ for $n\ge4$ of Nikiforov (the accepted partial claim Nikiforov 2005), the classical case $n=3$, the cases $n=4,5,6$ reported settled by the introductions of those papers (sources not held), and the case $n=7$ (Chen, Cheng and Zhang, European J. Combin. 29 (2008)) and the lengths $k=8$, $9$ and $10\le k\le15$ for $n=8$ (papers of 2007--2023), reported settled by the literature list of the 2026 manuscript below (sources not held), reduce it to the pairs with $8\le n<n_0(C)$ and $n\le k\le\min\{4n+1,\lceil C\log n/\log\log n\rceil-1\}$, less those $n=8$ cases (for $n=8$ the lengths $16\le k\le33$ below the threshold remain). The extent of that set is unknown because $C$ is not explicit; this is the finite check that the site's label DECIDABLE refers to. A manuscript claiming to close it in full is recorded on the pending claim page OpenAI 2026.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Acceptance. Refereed: Int. Math. Res. Not. IMRN 2021, no. 1, 275--300, doi:10.1093/imrn/rnz119, published online 10 July 2019 (the site's reference gives the pages as 277--302), the refereed evidence; first posted as arXiv:1807.06376v1 on 17 July 2018, the date this page is named by, the only arXiv version. The statement is cited from the arXiv preprint; the journal text is not compared. The site's curator, T. F. Bloom, labels the problem DECIDABLE and credits this paper in the page's commentary with the identity for all $k\ge C\log n/\log\log n$ and so with the conjecture for every large $n$; but the label DECIDABLE, defined on the site as resolved up to a finite check, settles neither the problem nor a declared part of it, since a finite check of unknown extent remains, so the curator's credit is not reviewed evidence and the claim is accepted on the refereed publication alone. The one comment of the site's discussion thread (1 September 2025) describes the problem as reduced to a finitary question and still open.

Read depth. Theorem 1.1 (arXiv:1807.06376v1, p. 2) and the remark on the constant (p. 16) are checked clause by clause; the proof is not checked, and nothing is independently reviewed in this corpus.