并封闭集猜测的两个加强形式

Two Stronger Versions of the Union-closed Sets Conjecture

并封闭集猜测(即Frankl猜测)即对于一个由有限个有限集组成的对并运算封闭的集族,如果这个集族至少包含一个非空集合,则存在一个元素至少在这个集族一半的集合里.定义M_(n)={1,2,…,n},F■2^(mn)={A:A■mn}满足∪_(A∈F)A=M_(n).假定F对并运算封闭,且Φ∈F.对任意k=1,2,…,n,定义Mk=(A∈2^(mn):|A|=k)及T(F)=inf{1≤k≤n:F∩M_(k)≠Φ}.基于T(F),本文引入Frankl猜测的两个加强形式(为叙述方便用S_(1)形式和S_(2)形式来表示).S_(1)形式说的是如果n≥2且T(F)=k∈{2,3,…,n},则存在M_(n)中的k个元素,它们都至少在F的一半的集合里.关于S_(1)形式,本文给出了部分结果.详细地说,本文证明了S_(1)形式当n≥2且T(F)=n,n-1,n-2时成立,当2≤n≤5时成立.另外,本文还提出了3个相关问题.

The union-closed sets conjecture(i.e.,Frankl’s conjecture)says that for any finite union-closed family of finite sets,other than the family consisting only of the empty set,there exists an element that belongs to at least half of the sets in the family.Let M_(n)={1,2,…,n}and■with∪_(A∈F) A=M_(n).Suppose that F is unionclosed andΦ∈F.For any k=1,2,…,n,denote■,and T(F)=inf{1≤k≤n:F∩M_(k)≠Φ}.By virtue of T(F),we introduce two stronger versions of Frankl’s conjecture(S_(1)-version and S_(2)-version for simplicity).S_(1)-version says that if n≥2 and T(F)=k∈{2,3,…,n},then there exist at least k elements in M_(n) which belong to at least half of the sets in F.We give some partial results on S_(1)-version.Precisely,we show that S_(1)-version holds for any n≥2 when T(F)=n,n-1,n-2,and S_(1)-version holds for 2≤n≤5.Three related questions are introduced.