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Cover-Free Family Through Biclique Covering

An (r,w;d) cover-free family (CFF) is a family of subsets of a finite set such that intersection of any r members of the family contains at least d points that are not in the union of w other members. The minimum number of points such that there exists an (r,w;d) − CFF with t blocks is denoted by N((r,w;d),t). In this paper we show that the value of N((r,w;d),t) is equal to d-biclique covering number of the graph It(r,w) whose vertices are the w- and r-subsets of a t-element set where two vertices are adjacent if their intersection is empty. We introduce some bounds for N((r,w;d),t). Also, we determine the exact value of N((r,w;d),t) for some values of d. Moreover, we show that there is a tight relationship between N((r,r;d),t) and the d-biclique covering number of the Kneser graph KG(t,r).