A Note on the Union-Closed Sets Conjecture

A union closed (UC) family \(\mathcal{A}\) is a finite family of sets such that the union of any two sets in \(\mathcal{A}\) is also in \(\mathcal{A}\). Peter Frankl conjectured that for every union closed family \(\mathcal{A}\), there exists some \(x\) contained in at least half the members of \(\mathcal{A}\). This is the union-closed sets conjecture.

An FC family is a UC family \(\mathcal{B}\) such that for every UC family \(\mathcal{A}\), if \(\mathcal{B} \subseteq \mathcal{A}\), then \(\mathcal{A}\) satisfies the union-closed sets conjecture. We give a heuristic method for identifying possible FC families, and apply it to families in \(\mathcal{P}(5)\) and \(\mathcal{P}(6)\).