发表机构
Delft Institute of Applied Mathematics, TU Delft; Mathematics Institute, University of Warwick; School of Mathematics and Statistics, University of Canterbury; Institute for Applied Mathematics, University of Bonn（代尔夫特应用数学研究所，代尔夫特理工大学; 华威大学数学研究所; 坎特伯雷大学数学与统计学院; 波恩大学应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了有限偏序集的维数上界可改进为$O(d \log d)$，其中$d$为可比图最大度，从而确认了Erdős等人下界的紧性。
AI 中文摘要
1986年，Füredi和Kahn证明了任意有限偏序集$P$的维数$\dim(P)$满足$\dim(P) = O(d \log^2 d)$，其中$d$是$P$的可比图的最大度。Scott和Wood最近将该界改进为$d \log^{1+o(1)} d$的形式。我们证明$\dim(P) = O(d \log d)$，从而确认了Erdős、Kierstead和Trotter的相应下界在隐常数意义下是紧的。
英文摘要
In 1986, Füredi and Kahn showed that the dimension $\dim(P)$ of any finite poset $P$ satisfies $\dim(P) = O(d \log^2 d)$, where $d$ is the maximum degree of the comparability graph of $P$. Scott and Wood more recently improved this bound to one of the form $d \log^{1+o(1)} d$. We show that $\dim(P) = O(d \log d)$, thus confirming that the corresponding lower bound of Erdős, Kierstead, and Trotter is tight up to the implicit constant.
Comments8 pages, 1 figure
