论文标题
对称分离的固定系统的纯度现象
The purity phenomenon for symmetric separated set-systems
论文作者
论文摘要
让$ n $成为一个积极的整数。 $ [n] = \ {1，\ ldots，n \} $的子集的集合$ \ cal s $被称为{\ it symmetric}，如果$ x \ in {\ cal s} $ in {\ cal s} $含义$ x^\ ast ast \ ast ast \ in {\ cal s} $ in { x \} $。我们表明，在三种类型的分离关系中的每一种中：{\ it强}，{\ it弱}和{\ it Chord}，以下“纯现象”发生：全包含在$ 2^{[n]} $中的最大最大对称分离集合。这些给出了众所周知的结果的“对称版本”，即通常强，弱和和弦分离的$ [n] $的纯度，而在较弱的分离的情况下，由于karpman对$ \ binom {[n]} $ N/2} $的纯度弱分离收集的纯度，这将扩大了最新结果。
Let $n$ be a positive integer. A collection $\cal S$ of subsets of $[n]=\{1,\ldots,n\}$ is called {\it symmetric} if $X\in {\cal S}$ implies $X^\ast\in {\cal S}$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$. We show that in each of the three types of separation relations: {\it strong}, {\it weak} and {\it chord} ones, the following "purity phenomenon" takes place: all inclusion-wise maximal symmetric separated collections in $2^{[n]}$ have the same cardinality. These give "symmetric versions" of well-known results on the purity of usual strongly, weakly and chord separated collections of subsets of $[n]$, and in the case of weak separation, this extends a recent result due to Karpman on the purity of symmetric weakly separated collections in $\binom{[n]}{n/2}$ for $n$ even.
