Completely separating systems of k-sets

AbstractDickson (1969) 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 ∈ S and j ∉ S. A collection of k-sets C is called a (n, k)-separator if, for each ordered pair (i, j) ∈ T × T with i ≠ j, there is a set S ∈ C which separates i from j. Let (n,k) denote the size of a smallest (n,k)-separator. For n⩾k(k − 1) we show that R(n,k) = [2n/k]. We also show that R(2m,2m−1)⩽2m and demonstrate various recursive relationships that are used to determine the exact values of R(n,k) for k⩽5