# Non trivial colouring of the edges of an infinite complete graph

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