Significant contribution to the Frankl's union-closed conjecture

Abstract

A celebrated unresolved conjecture of Peter Frankl states that every finite union-closed collection of sets ($B$), with non-empty universe, admits an abundant element. The best result in the literature states that if $|B|=n$, then there exists $x$ in the universe of $B$ with frequency at least $$\frac{n-1}{{\log }_{2}n}.$$ But $(n-1)/(n{\log }_{2}n)\to 0$ as $n\to \infty$.\\ In this paper, we show that there exists a constant $g>0$ such that for every $B$; there exists $x\in \text{U}(B)$ such that $$|B_x|\ge g|B|$$ where $B_x=\{A\in B:x\in A\}$ and $$\text{U}(B)=\bigcup _{A\in B}A.$$

comments show that the main theorem needs revision. So this form of the paper is incomplete!

References in corpus (1)

• Minimum density of union-closed families