On a problem of spencer

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.

This is a preview of subscription content, access via your institution.

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.

  2. [2]

    J. Spencer, Asymptotic Lower Bounds for Ramsey Functions,Discrete Math.,20(1977), 69–76.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shearer, J.B. On a problem of spencer. Combinatorica 5, 241–245 (1985). https://doi.org/10.1007/BF02579368

Download citation

AMS subject classification (1980)

  • 60 C 05
  • 05 C 99