Some new bounds for cover-free families through biclique covers

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 show that the value of N((r, w; d), t) is equal to the dbiclique covering number of the bipartite graph It(r, w) whose vertices are all wand r-subsets of a t-element set, where a w-subset is adjacent to an rsubset if their intersection is empty. Next, we introduce some new bounds for N((r, w; d), t). For instance, we show that for r ≥ w and r ≥ 2 N((r, w; 1), t) ≥ c ( r+w w+1 )