2022/01/31 by Milan Haiman, Haiman, Milan
Engineering · Mathematics · #06A07 #Advanced Combinatorial Mathematics #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2201.12952
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/κ, N] is bounded above by κ(logκ)1+o(1) and below by Ω((logκ/loglogκ)2). We improve the upper bound to O((log κ)3/(loglogκ)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.