The order dimension of divisibility

The Dushnik-Miller dimension of a partially-ordered set $P$ is the smallest $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$, is equal to ${(\log n)^2}(\log\log n)^{-\Theta(1)}$ as $n$ goes to infinity. We prove similar bounds for the $2$-dimension of divisibility in $\{1, \dotsc, n\}$, where the $2$-dimension of a poset $P$ is the smallest $d$ such that $P$ is isomorphic to a suborder of the subset lattice of $[d]$. We also prove an upper bound for the $2$-dimension of posets of bounded degree and show that the $2$-dimension of the divisibility poset on the set $(\alpha n, n]$ is $\Theta_\alpha(\log n)$ for $\alpha \in (0,1)$. At the end we pose several problems.