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.
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