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 c with probability $\frac{1}{2^c}$, for $c = 1, 2, \dots$ What is the probability that G contains an infinite monochromatic complete subgraph?
It is unclear for me if the answer should be $0, 1$, or something in between.