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An isoperimetric inequality for the Hamming cube and some consequences
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-07-20 , DOI: 10.1090/proc/15105 Jeff Kahn , Jinyoung Park
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-07-20 , DOI: 10.1090/proc/15105 Jeff Kahn , Jinyoung Park
Abstract:Our basic result, an isoperimetric inequality for Hamming cube 
, can be written: 
Here 
is uniform measure on 
(
); 
; and, for 
and 
, 
(where 
is the number of neighbors of 
in 
). This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of 
of size roughly 
, is a key step in showing that the number of maximal independent sets in 
is ![$ (1+o(1))2n\exp _2[2^{n-2}]$](images/img16.gif)
. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work.
中文翻译：
Hamming立方体的等距不等式和一些后果
摘要：我们的基本结果是汉明立方体的等距不等式可以写成： 这是对（）的统一度量；; 并且，对于和， （其中in的邻居数）。这意味着涉及边缘和顶点边界混合的不等式，以及相关的稳定性结果，并提出了一些更普遍的可能性。一个应用中，对于该组连接的两个不相交的子集的边缘的稳定性结果大小的大致，是在示出的在极大独立组的数量的关键步骤是。这种渐近陈述，其证据将单独出现，是本研究的最初动机。
更新日期：2020-08-20
, can be written: is uniform measure on (); ; and, for and , is the number of neighbors of in ). This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of of size roughly , is a key step in showing that the number of maximal independent sets in is . This asymptotic statement, whose proof will appear separately, was the original motivation for the present work. - [1]
中文翻译：
Hamming立方体的等距不等式和一些后果
摘要：我们的基本结果是汉明立方体的等距不等式
可以写成： 是对（）的统一度量；; 并且，对于和， 的邻居数）。这意味着涉及边缘和顶点边界混合的不等式，以及相关的稳定性结果，并提出了一些更普遍的可能性。一个应用中，对于该组连接的两个不相交的子集的边缘的稳定性结果大小的大致，是在示出的在极大独立组的数量的关键步骤是。这种渐近陈述，其证据将单独出现，是本研究的最初动机。 - [1]
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