Improved bounds for the dimension of divisibility

TL;DR

This paper establishes a tighter upper bound on the dimension of the divisibility order on integers up to n, refining previous bounds by analyzing connections with cover-free families and improving asymptotic estimates.

Contribution

It provides an improved asymptotic upper bound for the dimension of divisibility orders, refining previous results and connecting combinatorial structures with order dimension.

Findings

Upper bound on divisibility order dimension: C(log n)^2 (log log n)^{-2} log log log n

Refinement of Furedi and Kahn's bound

Connection established between order dimension and r-cover-free families

Abstract

The dimension of a partially-ordered set $P$ is the smallest integer $d$ 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 $\{1, \dotsc, n\}$ is bounded above by $C(\log n)^2 (\log \log n)^{-2} \log \log \log n$ as $n$ goes to infinity. This improves a recent result by Lewis and the first author, who showed an upper bound of $C(\log n)^2 (\log \log n)^{-1}$ and a lower bound of $c(\log n)^2 (\log \log n)^{-2}$, asymptotically. To obtain these bounds, we provide a refinement of a bound of F\"uredi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of $r$-cover-free families.