Suppose that $G=(V,E)$ is an infinite graph with finite degrees where every vertex is black with probability $p$.
The black subgraph is a subgraph whose vertices are black and the edges are edges between black vertices.
I am trying to find if:
The number of connected components is constant with probability 1.
The event that there are infinitely many connected components is a $0-1$ event.
Will the answers change if we allow infinite degrees?
It's pretty clear that kolmogorov 0-1 law and borel-cantelli lemmas will play a part here... However, I can't figure out the right way to use them.