Minimal Weight in Union-Closed Families

Let Ω be a finite set and let S ⊆ P(Ω) be a set system on Ω. For x ∈ Ω, we denote by dS(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then there is some x ∈ Ω with dS(x) ≥ 12 |S|. We consider a related question. Define the weight of a family S to be w(S) := ∑ A∈S |A|. Suppose S is union-closed. How small can w(S) be? Reimer showed w(S) ≥ 1 2 |S| log2 |S|, and that this inequality is tight. In this paper we show how Reimer’s bound may be improved if we have some additional information about the domain Ω of S: if S separates the points of its domain, then w(S) ≥ (|Ω| 2 )