The problem is this (recently asked about here):
A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat line up single file line. Each prisoner faces away from the beginning of the line, and each prisoner can see all the hats in front of him, and none of the hats behind. Starting from the beginning of the line, each prisoner must correctly identify the color of his hat or he is killed on the spot. The prisoners have a chance to meet and confer beforehand, but once in line, no prisoner can hear what the other prisoners say. The question is, is there a way to ensure that only finitely many prisoners are killed?
The standard solution depends on the Axiom of Choice, as discussed in this previous question.
However, the previous question only explains that that particular strategy requires the Axiom of Choice. It still seems at least conceivable that there could be a completely different strategy that works and doesn't need Choice.
Thus: Is it known to be consistent with ZF that there is no strategy for the prisoners?
(Bonus question: If "yes", then is this also true in the variant where the prisoners can hear the answers of lower-numbered prisoners?)