Two Stronger Versions of the Union-closed Sets Conjecture

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 this paper, we introduce two stronger versions of Frankl’s conjecture and give a partial proof. Three related questions are introduced.