Lower bounds for cover-free families

Let F be a set of blocks of a t-set X. A pair (X, F) is called an (w, r)-cover-free family ((w, r)–CFF) provided that, the intersection of any w blocks in F is not contained in the union of any other r blocks in F. We give new asymptotic lower bounds for the number of minimum points t in a (w, r)-CFF when w ≤ r = |F|^ε for some constant ε ≥ 1/2.