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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. For $r\ge3$ this answers Problem 55, which asks for any non-trivial bound on the growth of an $r$-Ramsey complete set: the construction is such a bound, since before it no $r$-Ramsey complete sequence with $|A\cap[n]|=n^{o(1)}$ was known even for $r=3$, and the lower bound adds the factor $r$ to the two-color bound $c\log^2n$, the counting form of Burr and Erdős's Theorem 2a (stated in their paper without proof), which carries over to every $r\ge2$ because an $r$-Ramsey complete sequence is $2$-Ramsey complete (refine any two-class partition into $r$ classes). The theorem thus determines the sparsest possible growth for every number of colors up to an absolute constant factor. The paper identifies the problem as the one for which Erdős offered a prize and says its first theorem solves it together with the two-color question, which is Problem 54. Adding the integers below the paper's threshold $n(A)$ makes the constructed sequence entirely $r$-Ramsey complete.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem SOLVED and credits the solution to the paper in the problem's commentary, which records the construction for every $r\ge2$ and the matching lower bound (accessed 2026-09-17; the page shows no last-edited date); 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 2026-09-17, with no journal version found (Crossref bibliographic query of the same date). The eight works citing the paper in the Semantic Scholar record of 2026-09-17 concern subset sums and knapsack algorithms and dispute nothing. Read depth: this page rests on the statement of Theorem 1.1 and the paragraphs around it, not on the proof, which the paper builds on its density Lemma 2.8; nothing here is independent review.

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