A Generalization of Cover Free Families
  An (r;w; d) -cover-free family (CFF) is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum number of elements for which there exists an (r;w; d)-CFF with t blocks is denoted by N((r;w; d); t). In this paper, we determine the exact value of N((r;w; d); t) for some special parameters. Also, we present two constructions for (2; 1; d)- CFF and (2; 2; d)-CFF which improve the existing constructions. Moreover, we introduce a generalization of cover-free families which is motivated by an application of CFF in the key predistribution schemes. Also, we investigate some properties and bounds on the parameters of this generalization. 
Mehdi Azadi Motlagh and Farokhlagha Moazami
math.CO