A Second Approximation To The Boundary Function On Union-Closed Collections

It is known that the boundary function \(\alpha\) on union-closed collections containing \(n\) sets has property \(\alpha(n) \leq \alpha(n)\), where \(\alpha(n)\) is Conway’s sequence. Herein a function \(f\) is defined on the positive integers and it is shown that for each value of \(n > 1\) a union-closed collection of \(n\) sets can be constructed with greatest element frequency \(\beta(n)\) and hence \(\alpha(n) \leq \beta(n)\); the inequality \(\beta(n) \leq \alpha(n)\) is proven for \(n \geq 1\) and so \(f\) is a closer approximation than \(\alpha\) to the boundary function \(\alpha\). It is also shown that \(\beta(n) \geq \frac{n}{2}\), thus incidentally providing an alternative proof to that of Mallows, that \(\alpha(n) \geq \frac{n}{2}\) for \(n \geq 1\).