Random covering designs

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A t - (n, k, λ) covering design (n ≥ k > t ≥ 2) consists of a collection of k-element subsets (blocks) of an n-element set script x sign such that each t-element subset of script x sign occurs in at least λ blocks. Let λ = 1 and k ≤ 2t - 1. Consider a randomly selected collection ℬ of blocks; |ℬ| = φ(n). We use the correlation inequalities of Janson to show that ℬ exhibits a rather sharp threshold behaviour, in the sense that the probability that it constitutes a t - (n, k, 1) covering design is, asymptotically, zero or one -according as φ(n) = {(nt)/(kt)}(log(nt) - w(n)) or φ(n) = {(nt)/(kt)}(log(nt) + w(n)), where w(n) → ∞ is arbitrary. We then use the Stein-Chen method of Poisson approximation to show that the restrictive condition k ≤ 2t - 1 in the above result can be dispensed with. More generaly, we prove that if each block is independently "selected" with a certain probability p, the distribution of the number W of uncovered t sets can be approximated by that of a Poisson random variable provided that E |ℬ| ≥ {(nt)/(kt)}[(t - 1) log n + log log n + an], where an → ∞ at an arbitrarily slow rate. © 1996 Academic Press, Inc.

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Journal of Combinatorial Theory. Series A