Small Sets in Union-Closed Families

Our aim in this note is to show that, for any $ϵ>0$, there exists a union-closed family $\mathcal{F}$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $ϵ$ of the sets in $\mathcal{F}$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{{\log }_{2}k}{2k}$ of the sets in $\mathcal{F}$. We also give explicit examples of union-closed families containing `small’ sets for which we have been unable to verify the Union-Closed Conjecture.