The number of 4-colorings of the Hamming cube

Let Q d be the d -dimensional hypercube and N = 2 d . We prove that the number of (proper) 4-colorings of Q d is asymptotically 6e2 N , as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980’s.