Some new bounds for cover-free families through biclique covers

An ( r , w ; d ) cover-free family ( C F F ) is a family of subsets of a finite set X 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 size of a set X for which there exists an ( r , w ; d ) − C F F with t blocks is denoted by N ( ( r , w ; d ) , t ) . s paper, we show that the value of N ( ( r , w ; d ) , t ) is equal to the d -biclique covering number of the bipartite graph I t ( r , w ) whose vertices are all w - and r -subsets of a t -element set, where a w -subset is adjacent to an r -subset if their intersection is empty. Next, we provide some new bounds for N ( ( r , w ; d ) , t ) . In particular, we show that for r ≥ w and r ≥ 2 N ( ( r , w ; 1 ) , t ) ≥ c r + w w + 1 + r + w − 1 w + 1 + 3 r + w − 4 w − 2 log r log ( t − w + 1 ) , where c is approximately 1 2 . Also, we determine the exact value of N ( ( r , w ; d ) , t ) for t ≤ r + w + r w and also for some values of d . Finally, we show that N ( ( 1 , 1 ; d ) , 4 d − 1 ) = 4 d − 1 if and only if there exists a Hadamard matrix of order 4 d .