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Claim. For every graph $G$ with $m$ edges and no isolated vertices,

$$r(G)\le2^{250\sqrt m},$$

where $r(G)$ is the least $N$ such that every two-coloring of the edges of $K_N$ contains a monochromatic copy of $G$. This is Theorem 1.1 of the library's source card, and it is the statement of Problem 546 with the constant $C=250$ in the exponent, so the claim settles the whole question. The paper notes that the bound is best possible up to the constant in the exponent: a complete graph with $m$ edges has Ramsey number at least $2^{\sqrt{m/2}}$ by Erdős's 1947 bound. The earlier refereed progress of Alon, Krivelevich and Sudakov, the bipartite case with $2^{16\sqrt m+1}$ and the general bound $2^{7\sqrt m\log_2m}$ for large $m$ (Combin. Probab. Comput. 12 (2003), Theorems 5.2 and 5.3), is recorded on its own partial claim page, Alon, Krivelevich and Sudakov 2003 (the bipartite case; the general bound settles no instance). The preprint arXiv:1002.0095v1 (30 January 2010) is the only arXiv version, and its date names this page.

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

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem PROVED and credits the theorem to Sudakov in the problem's commentary (page last edited 18 November 2025; accessed), an acceptance independent of the author; the community database records the problem as proved. Refereed: Advances in Mathematics 227 (2011), no. 1, 601--609 (the Crossref record, issued May 2011). The journal text is not held and has not been compared with the preprint, so locators are preprint pages. The twenty-eight citing records listed by Semantic Scholar include, by title, no correction or dispute. A Lean development in Boris Alexeev's repository declares itself a formalization of this theorem. It names Sudakov as informal author and Codex and GPT-5.6 Sol as formal authors, and it proves the bound with 65536 in place of 250. This corpus has not built it, so it gives no formalized evidence.

Read depth. Claims checked: Theorem 1.1 and the two introductory remarks around it on p. 2 of the preprint, clause by clause; the proof (Section 3, pp. 6--7, an embedding argument built on monochromatic pairs in the manner of the Erdős--Szekeres and Erdős--Szemerédi arguments) is outside this page's basis, and nothing is independently reviewed in this corpus. The analog for three or more colors is a separate open question, not this problem's.