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On a problem of Spencer. (English) Zbl 0587.60012
Let $$X_ 1,...,X_ n$$ be events in a probability space. Let $$\rho_ i$$ be the probability that $$X_ i$$ occurs. Let $$\rho$$ be the probability that none of the $$X_ i$$ occur. Let G be a graph on [n] so that for $$1\leq i\leq n$$ $$X_ i$$ is independent of $$\{X_ j| (i,j)\not\in G\}$$. Let f(d) be the sup of those x such that if $$\rho_ 1,...,\rho_ n\leq x$$ and G has maximum degree $$\leq d$$ then $$\rho >0$$. We show $$f(1)=1/2$$, $$f(d)=(d-1)^{d-1}d^{-d}$$ for $$d\geq 2$$. Hence $$\lim_{d\to \infty}df(d)=1/e$$. This answers a question posed by J. Spencer in Discrete Math. 20(1977), 69-76 (1978; Zbl 0375.05033). We also find a sharp bound for $$\rho$$ in terms of the $$\rho_ i$$ and G.

##### MSC:
 60C05 Combinatorial probability 05C99 Graph theory
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##### References:
  P. Erdos 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.  J. Spencer, Asymptotic Lower Bounds for Ramsey Functions,Discrete Math.,20(1977), 69–76. · Zbl 0375.05033 · doi:10.1016/0012-365X(77)90044-9
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