Small letters denote non-negative integers and capital letters denote sets. The set theoretic difference of A and B is A − B. The empty set is ø and |A| denotes the cardinal of A. We say that A is an immediate extension of B if B ⊂ A and |A − B| = 1. If C is a set of sets we put C(n) = {X: X ∈ C, |X| = n}. A set of sets C, has the property P(k, n) if (i) ø ∈ C and (ii) if X ∈ C(i) and i < n, then C(i + 1) contains at least k immediate extensions of X. We assume throughout that k ≥ 1.