论文标题
可划分的顺序维度
The order dimension of divisibility
论文作者
论文摘要
部分订购的套装$ p $的dushnik-miller尺寸是最小的$ 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 \} $上划分的尺寸等于$ {（\ log n）^2}（\ log \ log \ log \ log \ log n）^{ - θ（1）} $ as $ n $。 对于$ \ {1，\ dotsc，n \} $，我们证明了$ 2 $ dibimensible的$ 2 $数量的界限，其中$ 2 $ - poset $ p $的$ 2 $ dimension是最小的$ d $，因此$ p $是$ p $是对$ [d] $ $ [d] $的子集的子集的同构成的。我们还证明了一个上限的上限，以$ 2 $的限制限制限制的界限，并表明$ 2 $ - $ 2 $数的划分poset在集合$（αn，n，n] $中为$θ_α（\ log log n）$ for $α\ for $α\ in（0,1）$。最后，我们最终要解决几个问题。
The Dushnik-Miller dimension of a partially-ordered set $P$ is the smallest $d$ 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 equal to ${(\log n)^2}(\log\log n)^{-Θ(1)}$ as $n$ goes to infinity. We prove similar bounds for the $2$-dimension of divisibility in $\{1, \dotsc, n\}$, where the $2$-dimension of a poset $P$ is the smallest $d$ such that $P$ is isomorphic to a suborder of the subset lattice of $[d]$. We also prove an upper bound for the $2$-dimension of posets of bounded degree and show that the $2$-dimension of the divisibility poset on the set $(αn, n]$ is $Θ_α(\log n)$ for $α\in (0,1)$. At the end we pose several problems.
