Improved bounds for the dimension of divisibility

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, ... , 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 Furedi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of r-cover-free families. (c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).