Let N be a finite set, let p ∈ (0, 1), and let Np denote a random binomial subset of N where every element of N is taken to belong to the subset independently with probability p. This defines a product measure μp on the power set of N , where μp(A) := Pr[Np ∈ A] for A ⊆ 2N . In this paper we study monotone (upward-closed) families A for which all minimal sets in A have size at most k, for some ...