Lower Bounds for Cover-Free Families

Abstract

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