A generalization of Completely Separating Systems

A Completely Separating System (CSS) C on [ n ] is a collection of blocks of  [ n ] such that for any pair of distinct points x , y ∈ [ n ] , there exist blocks A , B ∈ C such that x ∈ A − B and y ∈ B − A . One possible generalization of CSSs are r -CSSs. Let T be a subset of 2 [ n ] , the power set of [ n ] . A point i ∈ [ n ] is called r -separable if for every r -subset S ⊆ [ n ] − { i } there exists a block T ∈ T with i ∈ T and with the property that S is disjoint from  T . If every point i ∈ [ n ] is r -separable, then T is an r -CSS (or r - ( n ) CSS). Furthermore, if T is a collection of k -blocks, then T is an r - ( n , k ) CSS. In this paper we offer some general results, analyze especially the case r = 2 with the additional condition that k ≤ 5 , present a construction using Latin squares, and mention some open problems.