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

TL;DR

This paper investigates the S-Frankl conjecture for union-closed families of sets, aiming to prove it for the case when the number of elements is six, building on previous results for up to five elements.

Contribution

It extends the proof of the S-Frankl conjecture to the case of six elements, providing a significant step forward in understanding the conjecture.

Findings

Proves the S-Frankl conjecture for n=6 elements

Builds on previous proofs for n≤5

Splits the proof into two parts due to length

Abstract

The union-closed sets conjecture (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. In [3], a stronger version of Frankl's conjecture (S-Frankl conjecture for short) was introduced and a partial proof was given. In particular, it was proved in \cite{CH17} that S-Frankl conjecture holds when $n\leq 5$, where $n$ is the number of all the elements in the family of sets. Now, we want to prove that it holds when $n=6$. Since the paper is very long, we split it into two parts. This is the first part.