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Conlon, Fox and Pham's Theorem 1.1 (arXiv:2104.14766v1, p. 3) fixes two absolute constants $C$ and $c>0$ that work for every number of colors $r\ge2$: some $r$-Ramsey complete sequence $A$ has at most $Cr\log^2n$ terms up to $n$ for every $n$, while a sequence with at most $cr\log^2n$ terms up to $n$ for every large $n$ is never $r$-Ramsey complete. At $r=2$ this answers Problem 54: the constructed sequence has $|A\cap[n]|\le2C\log^2n$ for all $n$, which replaces the cube of the logarithm in the site's second display by its square, and the lower bound matches the site's first display up to the constant, so the sparsest Ramsey $2$-complete sequence has counting function of order $\log^2n$ and only the constant factor remains. The paper identifies the problem as the Burr--Erdős question carrying Erdős's prize and says its first theorem solves it together with the question for $r\ge3$, which is Problem 55. Adding the integers below the paper's threshold $n(A)$ makes the constructed sequence entirely Ramsey $2$-complete with the same bound up to an additive constant.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem SOLVED and credits the resolution to the paper in the problem's commentary, which records the constructed Ramsey $2$-complete sequence with counting function $\ll(\log N)^2$ (page last edited 28 October 2025, accessed 2026-09-18); the discussion thread and the proof-claim tab are empty. The curator is independent of the authors. Not refereed: the paper is an arXiv preprint, version 1 of 30 April 2021 and the only version on the listing on 2026-09-18, with no journal version found (Crossref bibliographic query of the same date). The authors' own refereed 2022 Mathematika paper uses Theorem 6.1 of the preprint as its Theorem 3, which shows the authors' reliance on the preprint, not review by others. Read depth: this page rests on the statement of Theorem 1.1 and the paragraphs around it, not on the proof (Lemma 2.8 and Section 2); nothing here is independent review.

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