Minimal completely separating systems of k-sets

Let n and k be fixed positive integers. A collection C of k-subsets of [n] is a completely separating system if for all distinct i,j in [n] there is a set S in C that contains i but not j. Let R(n,k) denote the minimum size of a completely separating system of k-sets.

We determine, among other things, R(n,k) asymptotically when n=k1+o(1) and n/k goes to infinity. Several problems remain open.