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?

$\begingroup$ 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? $\endgroup$ – Tom Church Oct 28 '09 at 5:28

$\begingroup$ Well the infinite Ramsey theorem holds if you use only finitely many colours, but there is no such assumption here. Does that help? $\endgroup$ – Randomblue Oct 28 '09 at 10:29
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?

$\begingroup$ I didn't know the definition of a tail event, nor did I know Kolmogorow's 01 law. But yeah, it's seems you are right... Disappointing. $\endgroup$ – Randomblue Oct 27 '09 at 18:28