  Lionel Levine's hat challenge has $t$ players, each with a (very large, or
infinite) stack of hats on their head, each hat independently colored at random
black or white. The players are allowed to coordinate before the random colors
are chosen, but not after. Each player sees all hats except for those on her
own head. They then proceed to simultaneously try and each pick a black hat
from their respective stacks. They are proclaimed successful only if they are
all correct. Levine's conjecture was the success probability tends to zero when
the number of players grows. We prove that this success probability is strictly
decreasing in the number of players, and present some connections to questions
in graph theory.
