Minimal completely separating systems of k-sets

Let n and k be fixed positive integers. A collection s of lambda -sets of [ n] is a completely separating system if, for all distinct i, j is an elemen t of [n], there is an S is an element of s for which i is an element of S a nd j is not an element of S. Let R(n, k) denote the minimum size of such a s. Our results include showing that if n(k) is a sequence with lambda much less than n(k) much less than lambda (1+t) for every epsilon > 0, then R(n(lambda), k) - min{t: n(k) less than or equal to (1 / [kt n(lambda)]}. (C) 2001 Academic Press.