On the Frankl's union-closed conjecture

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.