Lower Bounds for Cover-Free Families

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 $w\le r=|{\cal F}|^ε$ for some constant $ε\ge 1/2$.

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