The Dimension of Divisibility Orders and Multiset Posets

TL;DR

This paper improves bounds on the dimension of divisibility orders and multiset posets, providing tighter estimates and extending results to polynomial divisibility orders.

Contribution

It refines the upper bound on the dimension of divisibility orders and extends the analysis to posets of multisets and polynomials.

Findings

Improved upper bound on divisibility order dimension to O((log κ)^3 / (log log κ)^2)

Established bounds for posets of multisets ordered by inclusion

Extended divisibility order analysis to polynomial divisibility orders

Abstract

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 $[N/\kappa, N]$ is bounded above by $\kappa (\log\kappa)^{1+o(1)}$ and below by $\Omega((\log\kappa/\log\log\kappa)^2)$. 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.