Random covering designs

A t - (n, k, lambda) covering design (n greater than or equal to k > t greater than or equal to 2) consists of a collection of k-element subsets (blocks) of an ii-element set H such that each t-element subset of H occurs in at least lambda blocks. Let lambda = 1 and k less than or equal to 2t - 1. Consider a randomly selected collection B of blocks; \B\ = phi(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, l) covering design is, asymptotically, zero or one-according [GRAPHICS] where omega(n) --> infinity is arbitrary. We then use the Stein-Chen method of Poisson approximation to show that the restrictive condition k less than or equal to 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 [GRAPHICS] where a(n) --> infinity at an arbitrarily slow rate. (C) 1996 Academic Press, Inc.