S_1-Frankl 猜想的6元情形（I）

The 6-element case of S_1-Frankl conjecture (I)

摘要:

并封闭集猜想（又称Frankl猜想）即对于一个由有限集合构成的关于并运算封闭的集族，如果这个集族至少包含一个非空集合，那么存在一个元素包含在至少一半的集合里.最近，本文作者提出了Frankl猜想的两个加强版本（简称为S_1-Frankl猜想与S_2-Frankl猜想），并给出了部分证明.特别地，作者证明：如果n<=5，则S_1-Frankl猜想成立，其中n表示这个集族中所有元素的个数.本文及其姊妹文证明当n=6时结论也成立.这是证明的第一部分.

Abstract:

The union-closed sets conjecture (Frankl's conjecture) says that for any nite union-closed family of nite 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. Recently, two stronger versions of the Frankl's conjecture (S_1-Frankl conjecture and S_2-Frankl conjecture for short) were introduced and partial proofs were given. In particular, it was proved that S_1-Frankl conjecture holds if n<=5, where n is the number of all the elements in the family of sets. In this paper and its sister paper, we prove that it holds if n = 6. This is the first part of the proof.