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

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.