§1. Let E = E(A) be the set of real numbers x ε (0, 1) whose regular continued fraction expansion

contains only partial denominators ai from a given set A of positive integers. For finite A the (Hausdorff-) dimensional numbers dim E have been studied by I. J. Good ‘2’ and T. W. Cusick ‘1’. C. A. Rogers ‘8’ introduced a natural probability measure on E. He showed that excluding sets with measure between 0 and 1 (in the strict sense) from E does not reduce the dimensionality more than excluding sets of measure zero, and that the minimal (“essential”) dimensions ess dim E arising in this way is smaller than dim E, at least for A = {1,…, r} when r = 2 or r is sufficiently large.

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. , if, and only if, ml/n → 1. When n is large and members of ℱ have cardinality k or k–1 we determine λ. For this we bound (ΔN)/N where ΔN is the shadow of Kruskal's k-cascade for the integer N. Roughly, if m ∼ N + ΔN, then λ ∼ kN with infinitely many cases of equality. A by-product is Theorem 7 of LYM posets.

The purpose of this paper is to prove the inequality of Theorem 1. The problem is due to B. Korenblum, who asked about it in connection with the characterization of the zero sets of functions analytic in the unit disc satisfying a growth condition. The problem was communicated to us by W. K. Hayman.