The Dimension of Divisibility Orders and Multiset Posets

The Dushnik–Miller dimension of a poset $P$ is the least $d$ for which $P$ can be embedded into a product of $d$ chains. Lewis and Souza showed that the dimension of the divisibility order on the interval of integers $[N\mathrm{/}\kappa ,N]$ is bounded above by $\kappa (\log \kappa {)}^{1+o(1)}$ and below by $\mathrm{\Omega }((\log \kappa \mathrm{/}\log \log \kappa {)}^2)$. We improve the upper bound to $O((\log \kappa)^3/(\log\log\kappa)^2).$ We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.