Dickson [On a problem concerning separating systems of a finite set, Journal of Combinatorial Theory, 7 (1969), 191--196.] introduced the notion of a completely separating set system. We study such systems with the additional constraint that each set in the system has the same size. Let T denote an n-set. We say that a subset S of T separates i from j if i 2 S and j 62 S. A collection of k-sets C is called a (n; k)-separator if, for each ordered pair (i; j) 2 T \Theta T with i 6 = j, there is a set S 2 C which separates i from j. Let R(n; k) denote the size of a smallest (n; k)-separator. For n k(k- 1) we show that R(n; k) = d2n=ke. We also show that R(2 m
