  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
$d$-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 introduce
some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$
and $r\geq 2$
  $$ N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose
w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the
well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we
determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we
show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of
order 4d.
