On homomorphisms from the Hamming cube to Z

Write F for the set of homomorphisms from {0, 1}(d) to Z which send (0) under bar to 0 (think of members of T as labellings of {0, 1}(d) in which adjacent strings get labels differing by exactly 1), and F-i for those which take on exactly i values. We give asymptotic formulae for |F| and \F-i\. In particular, we show that the probability that a uniformly chosen member f of T takes more than five values tends to 0 as d --> infinity. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constant b such that f a.s. takes at most b values. This in turn verified a conjecture of I. Benjamini et al., that for each t > 0, f a.s. takes at most td values. Determining |F| is equivalent both to counting the number of rank functions on the Boolean lattice 2([d]) (functions f: 2([d]) --> N satisfying f(circle divide) = 0 and f(A) less than or equal to f(AUx) less than or equal to f(A) + 1 for all A is an element of 2([d]) and x is an element of [d]) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from {0, 1}d to K-3, the complete graph on 3 vertices). Our proof uses the main lemma from Kahn's proof of constant range, together with some combinatorial approximation techniques introduced by A. Sapozhenko.