, Volume 5, Issue 3, pp 241–245

On a problem of spencer

  • J. B. Shearer

DOI: 10.1007/BF02579368

Cite this article as:
Shearer, J.B. Combinatorica (1985) 5: 241. doi:10.1007/BF02579368


LetX1, ...,Xn be events in a probability space. Let ϱi be the probabilityXi occurs. Let ϱ be the probability that none of theXi occur. LetG be a graph on [n] so that for 1 ≦i≦n Xi is independent of ≈Xj‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ1, ..., ϱnx andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1)d−1d−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 ϱi andG.

AMS subject classification (1980)

60 C 0505 C 99

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • J. B. Shearer
    • 1
  1. 1.Department of MathematicsU. C. BerkeleyBerkeleyUSA
  2. 2.IBM ResearchYorktown HeightsUSA