Significant contribution to the Frankl's union-closed conjecture

TL;DR

This paper proves a new lower bound on the frequency of an element in finite union-closed collections, advancing the understanding of Frankl's union-closed sets conjecture.

Contribution

It establishes the existence of a universal constant ensuring an element appears in a fixed proportion of the sets, improving previous bounds.

Findings

Existence of a constant g > 0 such that some element appears in at least g|B| sets.

Provides a bound independent of the size of B, unlike previous results.

Advances towards resolving 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 $$\texttt{U}(B)=\bigcup_{A\in B}A.$$