On a problem of spencer


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.

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  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.

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

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

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AMS subject classification (1980)

  • 60 C 05
  • 05 C 99