Families of Finite Sets in Which No Set Is Covered by the Union of Two Others

Let f k (n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f"(n) < f"(n + 1) < with equalities holding iff there exists a Steiner-system Y(t, 2t-1, n). The bounds are approximately best possible for k < 6 and of correct order of magnitude for k > 7, as well, even if the corresponding Steiner-systems do not exist. Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i .e., the nonuniform case). Let X be an n-element set. For an integer t, 0 < t < n we denote by (,) the collection of all the t-subsets of X, while 2 x denotes the set having all the different subsets of X as its elements. A family of subsets of X is just a subset of 2 X. We shall call it t-uniform if it is a subset of (X). By a Steiner-system Y = Y(t, k, n) we shall mean an Y c (k) such that for every A E (t) there is exactly one B E 5° with A c B. Obviously, we have ,Z(t, k, n)l (f)/(k). By [a]([b]) we shall denote the smallest (greatest) integer (not) exceeding a (b), respectively. We will use the Stirling formula, 1 .2. The Results THEOREM 1. Suppose .~k c (k) and there are no three distinct sets A, B, C E. k such that A c B U C. Let f k (n) denote the maximum of ,~r-k ~, subject to these constraints. Then we have 158