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The claim. For every $t$ and $\varepsilon>0$ there is $n_0(t,\varepsilon)$ such that for $n>n_0$

$$2^{c_t\log n/\log\log n}<f_t(n)<n^{3/4+\varepsilon}.$$

This is the Theorem, display (4), on p. 644 of P. Erdős, On a problem in elementary number theory and a combinatorial problem, Math. Comp. 18 (1964), no. 88, 644--646 (received 20 March 1964), paged as the theorem page of Erdős (1964). Erdős's $f_t(n)$, the least size that forces $t$ integers with pairwise the same greatest common divisor, is one more than the site's $f_r(N)$ of Problem 535. The upper bound comes from the Erdős–Rado sunflower bound applied to squarefree parts, the lower bound from an explicit product construction on the first $3k$ primes (p. 645). The page is named by the date the paper was received.

Covers. For every fixed $r\ge3$, $f_r(N)<N^{3/4+\varepsilon}$ for large $N$ and $f_r(N)>N^{c_r/\log\log N}$; 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 theorem and its proof are contained in the cited paper, whose card is linked above.