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Claim. Theorem 3 of the preprint states that for every integer $\ell\geq3$ and every graph $H$ with $m\geq1$ edges and no isolated vertices,

$$R(C_\ell,H)\leq(\ell-1)m+1\leq\ell m,$$

with no restriction relating $\ell$ and $m$. In the notation of Problem 569, where the cycle is $C_{2k+1}$, the theorem gives $c_k\leq2k+1$ for every $k\geq1$. The one-edge graph $K_2$ is eligible and $R(C_{2k+1},K_2)=2k+1$, so $c_k\geq2k+1$; hence the claim determines

$$c_k=2k+1$$

for every $k\geq1$, the case $k=1$ recovering the classical value $c_1=3$. The authors' summary on the site's proof-claim tab outlines the argument: an induction on the number of edges of $H$, lemmas under which a red path long enough inside a vertex's first or second red neighborhood forces a red $C_{2k+1}$, the selection of one vertex, a case analysis, and a final step that builds $H$ in blue when no red cycle has appeared. The preprint is carded at cambie_2026_general_bound_r_c_k_h, with the statement paged at Theorem 3; the text recorded there is arXiv v1 of 9 June 2026, seven pages.

Submission note. Posted to erdosproblems.com as a proof claim by Cambie Stijn and Freschi Andrea (account StijnC) on 25 July 2026, giving "none" as the AI used:

The extremal value for $c_k$ is obtained by the graph $H$ of minimum size; $H=K_2.$ So one proves $c_k=2k+1$ by induction, using a few lemmas (long red paths in the first or second red neighbourhood induce a red $C_{2k+1}$), a well chosen vertex, some case analysis and a conclusion by building a blue copy of $H$ in a few steps (if no red $C_{2k+1}$ appears).

Posted to the site's forum by Stijn Cambie on 10 June 2026:

The solution of this problem is now available at [CF26].

Scope. Full. The theorem covers every odd cycle length and every positive $m$, so together with the $K_2$ endpoint it answers the problem's question for every $k$; it does not assert that the bound $(\ell-1)m+1$ is attained for each individual $H$, which the problem does not ask.

Depends on. Goddard and Kleitman 1994 and Sidorenko 1993 ($R(C_3,H)\leq2m+1$, the preprint's Theorem 1), which the proof of Theorem 3 uses for $\ell=3$ and in its last step for every $\ell\geq7$; and Jayawardene 1999 for $\ell\in\{4,5,6\}$, through the preprint's Lemma 4 (p. 2), which for connected $H$ cites Theorems 4.1, 4.5 and 4.7 of the thesis together with four small Ramsey numbers from Radziszowski's dynamic survey. The case $k=2$, that is $\ell=5$, therefore comes through Lemma 4 from the thesis's Theorem 4.5, reported there as $R(C_5,H)\leq2m+2$ for every connected $H$ on at least four vertices (printed as an equality; the lemma uses the upper bound); the thesis is not held in this corpus, and its page records the result second-hand. The one-line endpoint deduction is made on the problem page and on the library's result page.

Standing. Claimed. The first author announced the preprint in the site's thread on 10 June 2026 as the problem's solution, and the proof claim of 25 July 2026 restates the argument; the tab states that listing a claim does not mean that anyone associated with the site has examined the proof. On 27 July 2026 a comment on the proof claim linked an automated audit, by the AI model the commenter names as GPT-5.6 Sol, alleging five repairable defects in the v1 proof, among them a mismatch between the auxiliary graph and the graph analyzed in the second-neighborhood step and an induction applied to a subgraph that may contain isolated vertices; the commenter wrote that the allegations had not been checked by hand. Searches found no reply, no repair and no later arXiv version; the arXiv record listed only v1 on 2026-10-07. The proof claim had two comments, both of 27 July 2026: the first author's remark that this may be the only full claim made without AI since the site's feature changed, and the comment linking the audit. No journal version was found (Crossref and OpenAlex, 2026-09-09). The site labels the problem OPEN (page last edited 18 January 2026) and the community database records it open. The dispute is unresolved and unverified either way; it is neither an acceptance nor a refutation, and the proof was not reconstructed or reviewed here, so no evidence kind is listed.