Lower bounds for cover-free families (Q2629487)

Summary: Let \(\mathcal F\) be a set of blocks of a \(t\)-set \(X\). A pair \((X,\mathcal F)\) is called an \((w,r)\)-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 \(w\leq r=|\mathcal F|^\epsilon\) for some constant \(\epsilon\geq 1/2\).