Clearly, it is possible to colour the edges of an infinite complete graph so that it does not contain any infinite monochromatic complete subgraph. Now what about the following?

Let $G$ be the complete graph with vertex set the positive integers. Each edge of $G$ is then coloured

cwith probability $\frac{1}{2^c}$, for $c = 1, 2, \dots$ What is the probability thatGcontains an infinite monochromatic complete subgraph?

It is unclear for me if the answer should be $0, 1$, or something in between.

infinitemonochromatic complete subgraph, that is. – Jonah Ostroff Oct 27 '09 at 17:16