Small Sets in Union-Closed Families

TL;DR

This paper constructs union-closed families with small minimal sets where elements are infrequently present, providing examples that challenge the Union-Closed Conjecture and exploring its boundaries.

Contribution

It demonstrates the existence of union-closed families with minimal sets having elements in a very small fraction of the sets, advancing understanding of the Union-Closed Conjecture.

Findings

Existence of union-closed families with minimal sets where elements appear in a fraction approaching zero.

Explicit examples of union-closed families with small sets that challenge the Union-Closed Conjecture.

Construction methods for families with minimal sets and low element frequency.

Abstract

Our aim in this note is to show that, for any $\epsilon>0$, 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 $\epsilon$ 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.