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The claim. For every $\varepsilon>0$ and $n\ge n_0(\varepsilon)$, $f(n)>(1-\varepsilon)n\log\log n/\log n$, where $f(n)$ is the largest size of a subset of $\{1,\ldots,n\}$ with no three members of pairwise the same least common multiple, the function of Problem 536. With $l=[n^{1/4}]$, the products $P_iP_{l+j}$ of the $i$-th prime ($i\le l$) with the primes $P_{l+j}\le n/P_i$ contain no three with pairwise the same least common multiple, and their number exceeds the bound. This is display (11) of H. L. Abbott and B. Gardner, An extremal problem in number theory, Canad. Math. Bull. 10 (1967), no. 2, 173--177 (received 17 November 1966), pp. 176--177, paged as display (11) of Abbott and Gardner (1967). The page is named by the date the paper was received.

Covers. The lower bound only; neither $f(N)=o(N)$ nor the order of $f(N)$ is settled.

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

Depends on. Nothing in this wiki: the bound and its proof are contained in the cited paper, whose card is linked above.