论文标题
布尔晶格的下属的局部维度
The local dimension of suborders of the Boolean lattice
论文作者
论文摘要
我们证明了布尔晶格的任何一对层的局部维度上的上限和下限，并表明$ n $维的boolean lattice的第一和中间层的局部维度是渐近的$ \ frac {n} {\ log_2 n} $ as $ n \ as $ n \ to uftty $。以前，所有已知的是$ω（n/\ log n）$的下限和$ n $的上限。 改善金，马丁，马萨报，舒尔，史密斯，乌泽尔和王的结果，我们还证明，$ n $ emlement poset的最大局部维度至少为$ \ left（\ frac {1} {1} {4} {4} {4} -O（1）-o（1）\ right）
We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that the local dimension of the first and middle layers of the $n$-dimensional Boolean lattice is asymptotically $\frac{n}{\log_2 n}$ as $n\to\infty$. Previously, all that was known was a lower bound of $Ω(n/\log n)$ and an upper bound of $n$. Improving a result of Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang, we also prove that that the maximum local dimension of an $n$-element poset is at least $\left(\frac{1}{4}-o(1)\right)\frac{n}{\log_2 n}$.
