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.



Infinite Ramsey theorem with infinitely many colours
Every countably infinite random graph is almost surely the Rado graph which contains all finite and countably infinite graphs as induced subsets. So each color class almost surely contains the Rado graph and hence a infinite monochromatic subgraph. See the following for more details here:


<http://en.wikipedia.org/wiki/Random_graph>


There are also links in the article to other articles including one on the Rado graph.


