TITLE: Non trivial colouring of the edges of an infinite complete graph
QUESTION [6 upvotes]: 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?

REPLY [9 votes]: 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?