Here is a problem that I can not solve. Any suggestion or hint would be helpful. If there is a history of this problem in the literature please let me know. 

Does there exists subsets $A\_1, A\_2, \cdots$ of $\mathbb{N}$ such that for any two (not necessary different) natural numbers $i,j$, $|A\_i\cap A\_j| = (i,j)$ ? (Where $(i,j)$ is the greatest common divisor of $i$ and $j$.) As joriki also pointed out, this implies that $|A\_k|=k$ for all $k$.