The epsilon-union-closed sets conjecture

Let $\mathcal{A}$ be a finite union-closed collection of sets containing at least one non-empty set, and let $|\mathcal{A}|$ denote its number of sets. The $\varepsilon$-union-closed sets conjecture. There exists $\varepsilon>0$ such that there is an element belonging to at least $\varepsilon\cdot |\mathcal{A}|$ of the sets in $\mathcal{A}$. This is a weakening of the union-closed sets conjecture, proposed because the original problem has proved difficult; the source provides no resolution of this weakened conjecture.