Random Covering Designs

Abstract

At−(n, k, λ) covering design (n⩾k>t⩾2) consists of a collection ofk-element subsets (blocks) of ann-element set X such that eacht-element subset of X occurs in at leastλblocks. Letλ=1 andk⩽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 at−(n, k, 1) covering design is, asymptotically, zero or one—according asφ(n)={(nt)/(kt)}(log(nt)−ω(n)) orφ(n)={(nt)/(kt)}(log(nt)+ω(n)), whereω(n)→∞ is arbitrary. We then use the Stein–Chen method of Poisson approximation to show that the restrictive conditionk⩽2t−1 in the above result can be dispensed with. More generaly, we prove that if each block is independently “selected” with a certain probabilityp, the distribution of the numberWof uncoveredtsets can be approximated by that of a Poisson random variable provided thatE|B|⩾{(nt)/(kt)}[(t−1)logn+loglogn+an], wherean→∞ at an arbitrarily slow rate.