arXiv 1206.31522012-06-14math.CO

关于汉明立方体到{ \bf Z}的同态

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

David Galvin

备注：: 27 pages. Appeared in Israel Journal of Mathematics in 2003

摘要：

对于从$\{0,1\}^d$到${\bf Z}$的同态的集合，将$\underline{0}$设为0（将${\cal F}$的成员视为$\{0,1\}^d$的标签，其中相邻字符串的标签恰好相差1），并将${\cal F}_i$用于那些恰好具有$i$值的成员。给出了$|{\cal F}|$和$|{\cal F}_i|$的渐近公式。特别是，我们表明，一个统一选择的成员${\bf f}$的${\cal F}$采取超过五个值的概率趋于0的$d \rightarrow \infty$。这就解决了J. Kahn的一个猜想。之前，Kahn已经证明存在恒定的$b$，使得${\bf f}$ a.s.最多取$b$值。这反过来证实了我的一个猜想。Benjamini { \em等}，对于每个$t > 0$，${\bf f}$ a.s.最多取$td$值。确定$|{\cal F}|$既等同于对布尔格$2^{[d]}$上的秩函数（对于所有$A \in 2^{[d]}$和$x \in [d]$，满足$f(\emptyset)=0$和$f(A) \leq f(A \cup x) \leq f(A)+1$的函数$f \colon 2^{[d]} \longrightarrow {\bf N}$）的数目进行计数，又等同于对离散立方体的适当3-着色的数目进行计数（即，从$\{0,1\}^d$到$K_3$的同态数，3个顶点上的完全图）。我们的证明使用了Kahn关于常值域证明的主要引理，并结合了A。萨波任科

英文摘要：

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.

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英文摘要：

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