Let n and k be xed positive integers. A collection C of k-sets of n] is a completely separating system if, for all distinct i; j 2 n], there is an S 2 C for which i 2 S and j 6 2 S. Let R(n; k) denote the minimum size of such a C. Our results include showing that if n k is a sequence with k n k k 1+ for every > 0, then