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 $$\frac{n-1}{\log_2n}.$$ But $(n-1)/(n\log_2n)\rightarrow 0$ as $n\rightarrow \infty$.\\ In this paper, we show that there exists a constant $g>0$ such that for every $B$; there exists $x\in \texttt{U}(B)$ such that $$|B_x|\geq g|B|$$ where $B_x=\{A\in B: x\in A\}$ and $$\texttt{U}(B)=\bigcup_{A\in B}A.$$