Improved bounds for the dimension of divisibility

The dimension of a partially-ordered set $P$ is the smallest integer $d$ 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 bounded above by $C(\log n)^2 (\log \log n)^{-2} \log \log \log n$ as $n$ goes to infinity. This improves a recent result by Lewis and the first author, who showed an upper bound of $C(\log n)^2 (\log \log n)^{-1}$ and a lower bound of $c(\log n)^2 (\log \log n)^{-2}$, asymptotically. To obtain these bounds, we provide a refinement of a bound of F\"uredi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of $r$-cover-free families.