Abstract
Let ${\cal F}$ be a set of blocks of a $t$-set $X$. A pair $(X,{\cal F})$ is called an $(w,r)$-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 $\epsilon\ge 1/2$.
Highlights
Let F be a set of blocks of a t-set X
The best known lower bound for N (n, (1, r)) is [2, 4, 7]
Assume the bound holds for some k w and every r that satisfies r (n + k − w) k+1
Summary
Let F be a set of blocks (subsets) of a t-set X. A pair (X, F) is called a (w, r)-cover-free family ((w, r)−CFF) provided that, for any w blocks A1, A2, . Let N (n, (w, r)) denote the minimum number of points |X| in any (w, r)-CFF having |F| = n blocks. The best known lower bound for N (n, (1, r)) is [2, 4, 7]. √ In this paper we give a new lower bound for (w, r)-CFF when r > n. We combine the two techniques used in [8, 6] and [1] to give the following asymptotic lower bound. Lnk+1 r log n for k−1 k (n + k − 1 − w) k r (n + k − w) k+1 and n.
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