  Let ${\cal F}$ be a set of blocks of a $t$-set $X$. $(X,{\cal F})$ is called
$(w,r)$-cover-free family ($(w,r)-$CFF) provided that, the intersection of any
$w$ blocks in ${\cal F}$ is not contained in the union of any other $r$ blocks
in ${\cal F}$. We give new asymptotic lower bounds for the number of minimum
points $t$ in a $(w,r)$-CFF when $w\le r=|{\cal F}|^\epsilon$ for some constant
$\epsilon\ge 1/2$.
