论文标题
改善了分裂性维度的界限
Improved bounds for the dimension of divisibility
论文作者
论文摘要
部分订购的套装$ p $的尺寸是最小的整数$ d $，因此可以将$ p $嵌入$ d $线性订单的产品中。我们证明，在间隔$ \ {1，\ dotsc，n \} $上的划分顺序的尺寸在上面由$ c（\ log n）^2（\ log \ log \ log \ log \ log n）^{ - 2} \ log \ log \ log \ log \ log \ log n $作为$ n $ as $ n $ as $ as $ a a $ a a $ as $ n $ as $ a in infinity。这改善了刘易斯和第一作者的最新结果，后者显示了$ c（\ log n）^2（\ log \ log \ log \ log n）^{ - 1} $的上限，以及$ c（\ log n）^2（\ log \ log \ log \ log \ log n）^{ - 2} $的下限。为了获得这些界限，我们提供了Füredi和Kahn的界限的改进，并利用了划分命令的维度与最大规模的最大尺寸为$ r $ cover-cover的家庭之间的联系。
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 Füredi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of $r$-cover-free families.
