On homomorphisms from the Hamming cube to {\bf Z}

TL;DR

This paper provides asymptotic formulas for the number of homomorphisms from the Hamming cube to integers, showing that most such functions take very few values as the dimension grows, confirming conjectures by Kahn and Benjamini et al.

Contribution

It establishes asymptotic counts for homomorphisms from the Hamming cube to Z and proves that such functions almost surely take at most five values as dimension increases, settling longstanding conjectures.

Findings

Probability of more than five values tends to zero as dimension grows

Number of homomorphisms relates to counting rank functions and 3-colorings

Confirms conjectures of Kahn and Benjamini et al.

Abstract

Write ${\cal F}$ for the set of homomorphisms from $\{0,1\}^d$ to ${\bf Z}$ which send $\underline{0}$ to 0 (think of members of ${\cal F}$ as labellings of $\{0,1\}^d$ in which adjacent strings get labels differing by exactly 1), and ${\cal F}_i$ for those which take on exactly $i$ values. We give asymptotic formulae for $|{\cal F}|$ and $|{\cal F}_i|$. In particular, we show that the probability that a uniformly chosen member ${\bf f}$ of ${\cal F}$ takes more than five values tends to 0 as $d \rightarrow \infty$. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constant $b$ such that ${\bf f}$ a.s. takes at most $b$ values. This in turn verified a conjecture of I. Benjamini {\em et al.}, that for each $t > 0$, ${\bf f}$ a.s. takes at most $td$ values. Determining $|{\cal F}|$ is equivalent both to counting the number of rank functions on the Boolean lattice $2^{[d]}$ (functions $f \colon 2^{[d]} \longrightarrow {\bf N}$ satisfying $f(\emptyset)=0$ and $f(A) \leq f(A \cup x) \leq f(A)+1$ for all $A \in 2^{[d]}$ and $x \in [d]$) 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.