Solved. Conlon, Fox and Pham's Theorem 1.1 determines the sparsest possible growth for every $r\ge2$ up to an absolute constant factor: there is an $r$-Ramsey complete $A$ with $|A\cap[n]|\le Cr\log^2n$ for all $n$, and no $A$ with $|A\cap[n]|\le cr\log^2n$ for all large $n$ is $r$-Ramsey complete. The status-defining source is an arXiv preprint of 2021; the site's curator accepted it and names it as the solution. The problem asks for bounds rather than for a proposition, so the catalog label is SOLVED and not PROVED. On the curator's acceptance the claim page records the result as accepted, with no refereed version, and the frontmatter standing is derived from it.