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.