Countably Infinitely Many Hats
Countably infinitely many mathematicians have been taken hostage. They are indexed by the positive integers (each mathematician knows their integer) and are standing in a line so that mathematician \(n\) can see each mathematician \(m\) for \(m>n\).
The mathematicians are each given a red or blue hat; they cannot see their own. Starting from the back (mathematician 1), each mathematician will guess their own hat color (one at a time). They can hear all previous answers. All mathematicians who guess correctly may leave after their turn. The mathematicians are not allowed to communicate extra information other than their guess (e.g. by gesturing or changing tone) on pain of death to all the participants - they must simply guess.
Beforehand, the mathematicians have agreed upon a strategy to minimize the number of mathematicians who might guess incorrectly. Ignoring problems like limited vision or finite memory capacity and assuming the axiom of choice holds, what is the greatest possible number of mathematicians that might guess incorrectly?