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 will also see similar results when for variant notions of dimension and when the divisibility order taken over various other sets of integers.
Based on joint work with David Lewis and also with Leo Versteegen.
