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Can you build a probabilistic scheme for colouring each edge (independently of all other edges) of the complete graph G on the positive integers such that the probability that G contains an infinite monochromatic complete subgraph is neither 0 nor 1?

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The question has been answered, but I'm still confused. Isn't Ramsey's theorem exactly the statement that there is always a infinite monochromatic complete subgraph, so that the probability would be identically 1? –  Tom Church Oct 28 '09 at 5:28
    
Well the infinite Ramsey theorem holds if you use only finitely many colours, but there is no such assumption here. Does that help? –  Randomblue Oct 28 '09 at 10:29
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up vote 7 down vote accepted

It seems like this would go against the Kolmogorov 0-1 law.. If we let Xi denote the coloring of all of the edges from i to integers larger than i, wouldn't the existence of an infinite monochromatic subgraph be a tail event?

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I didn't know the definition of a tail event, nor did I know Kolmogorow's 0-1 law. But yeah, it's seems you are right... Disappointing. –  Randomblue Oct 27 '09 at 18:28
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