FAMILIES OF FINITE SETS IN WHICH NO SET IS COVERED BY THE UNION OF r OTHERS'

Let f,(n, k) denote the maximum number of k-subsets of an n-set satisfying the condition in the title . It is proved that f,(n,r(t-1)+1+d)-(n t dl/ u lk t dl for n sufficiently large whenever d = 0,1 or d < r/2 t 2 withJh equality holding iff there exists a Steiner system S(t, r(t 1) + l, n d). The determination of f,(n, 2r) led us to a new generalization of BIRD (Definition 2 .4). Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family . 1 . Preliminaries Let X be an n-element set . For an integer k, 0-k < n we denote by (k) the collection of all the k-subsets of X, while 2' denotes the power set of X. A family of subsets of X is just a subset of 2X . It is called k-uniform if it is a subset of (k) . A Steiner system Y = S(t, k, n) is an Y C (k) such that for every T E (;`) there is exactly one B E Y with T C B . Obviously, holds. A C (k) is called a (t, k, n)-packing if I P fl P , I < t holds for every pair P, Y E 91. V . Rödl [10] proved that ISRAEL JOURNAL OF MATHEMATICS, Vol . 51, Nos. 1-2, 1985 FAMILIES OF FINITE SETS IN WHICH NO SET IS COVERED BY THE UNION OF r OTHERS'