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The claim. For every fixed $r\ge3$ and $\varepsilon>0$, $f_r(N)<N^{1/2+\varepsilon}$ for all large $N$, where $f_r(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no $r$ elements of pairwise the same greatest common divisor, as in Problem 535. The result is H. L. Abbott and D. Hanson, An extremal problem in number theory, Bull. London Math. Soc. 2 (1970), no. 3, 324--326. The paper is not held, and no review of it was found, so the result is stated in the form Erdős's 1973 survey gives for it, Section 4, printed p. 123 of Erdős (1973): "I proved $f_r(x)<x^{3/4+\varepsilon}$ (Erdős [1964a]). This was improved to $x^{1/2+\varepsilon}$ by Abbott and Hanson [1970]." The page is named by the paper's issue, November 1970 according to its Crossref record, with the first day of the month standing in for the unknown day.

Covers. The upper exponent $1/2$ only; the order of $f_r(N)$ is not determined.

Acceptance. Refereed: the journal publication. The site's commentary credits the result on a problem it labels OPEN, which is not acceptance.

Depends on. Nothing in this wiki: the result is the cited paper's, as the linked survey reports it.