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,…,n} is bounded above by C(logn)2(loglogn)−2logloglogn as n goes to infinity. This improves a recent result by Lewis and the first author, who showed an upper bound of C(logn)2(loglogn)−1 and a lower bound of c(logn)2(loglogn)−2, asymptotically. To obtain these bounds, we provide a refinement of a bound of Füredi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of r-cover-free families.