论文标题
联盟封锁的家庭的小型
Small Sets in Union-Closed Families
论文作者
论文摘要
我们在本说明中的目的是表明，对于任何$ε> 0 $，都存在一个联合封闭的家庭$ \ Mathcal f $，其中（唯一）最小的set $ s $，因此，$ s $的元素不属于$ \ \ \ \ \ \ \ \ m nathcal f $的$ s $的分数$ε$。更确切地说，我们举了一个示例，它的示例是一个最小的尺寸$ k $的联合锁定家庭，因此该集合中的任何元素都不属于$ \ \ \ mathcal f $ in $ \ mathcal f $的$（1+o（1））\ frac {\ frac {\ log_2 k} {2k} $。我们还提供了包含“小”集合的联盟锁定家庭的明确例子，我们无法验证工会关闭的猜想。
Our aim in this note is to show that, for any $ε>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $ε$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture.
