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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It seems like this would go against the Kolmogorov 01 law.. If we let X_{i} 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? 

