Two Stronger Versions of the Union-closed Sets Conjecture

TL;DR

This paper proposes two stronger variants of Frankl's union-closed sets conjecture, introduces related questions, and provides partial proofs, advancing understanding of the conjecture's validity.

Contribution

It introduces two new, stronger versions of the union-closed sets conjecture and offers partial proofs, along with three related open questions.

Findings

Proposed two stronger versions of the conjecture

Provided partial proofs for these versions

Introduced three related open questions

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