The success probability in Lionel Levine's hat problem is strictly decreasing with the number of players, and this is related to interesting questions regarding Hamming powers of Kneser graphs and independent sets in random subgraphs

Ehud Friedgut, Gil Kalai, Guy Kindler

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 playe…