New formulations of the union-closed sets conjecture

Abstract

The union-closed sets conjecture states that if a finite set \(\mathcal A\) of finite sets is union-closed and \(\mathcal A\neq \{ \varnothing\}\), then there exists an element in \(\displaystyle\cup_{A\in \mathcal A} A\) that belongs to at least half of the sets in \(\mathcal A\). We present three new formulations of the union-closed conjecture in terms of matrices, graphs, and hypergraphs.