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?



Non trivial colouring of the edges of an infinite complete graph
It seems like this would go against the [Kolmogorov 0-1 law.](http://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_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? 


