The order dimension of divisibility

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