On Kruskal's cascades and counting containments in a set of subsets

Let ℱ be a set of m subsets of X = {1,2,…, n }. We study the maximum number λ of containments Y ⊂ Z with Y, Z ∊ ℱ. Theorem 9. λ = ( 1 + o ( 1 ) ) (2 m ) , if, and only if , m l/n → 1. When n is large and members of ℱ have cardinality k or k–1 we determine λ. For this we bound (Δ N )/ N where Δ N is the shadow of Kruskal's k ‐cascade for the integer N . Roughly, if m ∼ N + Δ N , then λ ∼ kN with infinitely many cases of equality. A by‐product is Theorem 7 of LYM posets.