A beautiful puzzle, and I know the solution. I read it on an Italian riddle forum.
An evil witch kidnaps infinite (countable) dwarves. The witch tells the dwarves that the next day she will put them all facing the same direction and uniformly at random will place on each dwarf's head a hat that can be either white or black. So that the first one sees all the other dwarves. The second all but the first, etc, i.e. each dwarf will then see only the color of the dwarves' hats in front of him.
Each dwarf will have to guess their hat color, whoever guesses is free whoever misses dies. The dwarves have one night to be able to discuss a strategy. The main problem is that the dwarves are deaf! So they will not hear anything the other dwarves say. Can you come up with a strategy that can save all but finitely many dwarves?
Edit: My question is very different from Infinitely many dwarves wearing hats of 2 colours because if the dwarves are not deaf you can find a strategy for which you can save all the dwarves except the first one. However, it is true that the answer given (by theosza) in the other question works perfectly to my question. The difference is the fact that the answer of theosza is not the optimal solution to Infinitely many dwarves wearing hats of 2 colours, and the optimal solution of Infinitely many dwarves wearing hats of 2 colours is not applicable to the case where the dwarves are deaf.