, Volume 5, Issue 3, pp 241-245

On a problem of spencer

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Abstract

LetX 1, ...,X n be events in a probability space. Let ϱi be the probabilityX i occurs. Let ϱ be the probability that none of theX i occur. LetG be a graph on [n] so that for 1 ≦i≦n X i is independent of ≈X j ‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ1, ..., ϱ n x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1) d−1 d −d ford≧2. Hence \(\mathop {\lim }\limits_{d \to \infty } \) df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ i andG.