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Claim. In Some extremal problems in combinatorial number theory (Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press (1970), 123–133; library card erdos_1970_extremal_problems_combinatorial_number_theory), Erdős proves on p. 127, through displays (19) and (20), that for every $c>0$ and $x>x_0(c,k)$ a set of integers below $x$ whose reciprocal sum exceeds $c\log x$ contains $k$ members with pairwise the same least common multiple. The proof finds an integer $t$ with at least $k$ representations $t=a_ip$, $p$ prime: otherwise summing $1/(a_ip)$ over the set and the primes below $x$ bounds $(\sum1/a_i)\log\log x$ by a multiple of $k\log x$. So the function $f_k(N)$ of Problem 856 is $o(\log N)$, and the same counting gives $f_k(N)\ll_k\log N/\log\log N$, the bound the site's commentary states with this proof.

Covers. The upper bound $f_k(N)\ll_k\log N/\log\log N$. Not covered: the order of $f_k(N)$, for which Erdős asks on the same page how far the hypothesis can be weakened.

Standing. Claimed: the paper is a chapter of a dedication volume, not a journal, so it is not listed as refereed. The site's commentary credits the bound to Erdős on a problem it labels OPEN; that credit is commentary, not acceptance.

Depends on. No page of this wiki.