Combinatorica

, Volume 5, Issue 3, pp 241–245 | Cite as

On a problem of spencer

  • J. B. Shearer
Article

Abstract

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 05 05 C 99 

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References

  1. [1]
    P. Erdős andL. Lovász, Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions,Infinite and Finite Sets, Colloquia Mathematica Societatis János Bolyai, Keszthely (Hungary), 1973, 609–627.Google Scholar
  2. [2]
    J. Spencer, Asymptotic Lower Bounds for Ramsey Functions,Discrete Math.,20(1977), 69–76.CrossRefMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • J. B. Shearer
    • 1
  1. 1.Department of MathematicsU. C. BerkeleyBerkeleyUSA

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