Erdős problem 535

Let $r \geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all $r \geq 3$.