对称分离集系统的纯性现象
The purity phenomenon for symmetric separated set-systems
摘要：
令$n$为正整数。如果$X\in {\cal S}$隐含$X^\ast\in {\cal S}$，则$[n]=\{1,\ldots,n\}$的子集的集合$\cal S$被称为{ \it对称}，其中$X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$。我们证明了在{ \it strong}、{ \it weak}和{ \it chord}三种分离关系中，都发生了如下的“纯性现象”：$2^{[n]}$中所有包含式最大对称分离集合具有相同的基数.这些给“对称版本”的知名结果的纯度通常强，弱和弦分离集合的子集$[n]$，并在弱分离的情况下，这延长了最近的结果由于Karpman的纯度对称弱分离集合在$\binom{[n]}{n/2}$为$n$甚至。
英文摘要：
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.
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