On the Frankl's union-closed conjecture

TL;DR

This paper proves Frankl's union-closed conjecture, asserting that in any finite union-closed family of sets, there exists an element present in at least half of the sets, using induction and irreducible collections.

Contribution

The paper introduces the concept of irreducible collections and proves the conjecture for all such collections, advancing the understanding of union-closed families.

Findings

Proof of Frankl's union-closed conjecture for all finite collections.

Introduction of irreducible collections as a key concept.

Establishment that the conjecture holds for irreducible collections with non-empty universe.

Abstract

A celebrated unresolved conjecture of Peter Frankl states that every finite collection of sets, with finite universe, admits an abundant element. In this paper, we prove Frankl's union-closed conjecture(FC). We provide an induction proof based on a key result that every candidate collection, $\Omega$, admits an irreducible form. The concept of irreducible collections is one of the key contributions of this paper. We show that the conjecture is true if it holds for a class of irreducible finite collections. Then we show that the conjecture holds for all irreducible finite collections, with non-empty universe.