The order dimension of divisibility

Abstract

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,…,n}, is equal to (log⁡n)2(log⁡log⁡n)−Θ(1) as n goes to infinity.We prove similar bounds for the 2-dimension of divisibility in {1,…,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 (αn,n] is Θα(log⁡n) for α∈(0,1). At the end we pose several problems.