The Hamming graph $H(n,q)$ is defined on the vertex set $\{1,2,\ldots,q\}^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon proved that for any sequence $v_1,\ldots,v_b$ of $b=\lceil\frac n2\rceil$ vertices of $H(n,2)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. In this note, we prove that for any $q\geq 3$ and any sequence $v_1,\ldots,v_b$ of $b=\lfloor(1-\frac1q)n\rfloor$ vertices of $H(n,q)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. Alon used the Beck--Spencer Lemma which, in turn, was based on the floating variable method introduced by Beck and Fiala who studied combinatorial discrepancies. For our proof, we extend the Beck--Spencer Lemma by using a multicolor version of the floating variable method due to Doerr and Srivastav.

翻译：汉明图$H(n,q)$定义在顶点集$\{1,2,\ldots,q\}^n$上，当且仅当两个顶点恰有一个坐标不同时，它们相邻。阿隆证明了对于$H(n,2)$中任意$b=\lceil\frac n2\rceil$个顶点的序列$v_1,\ldots,v_b$，总存在一个顶点，其到$v_i$的距离至少为$b-i+1$（对所有$1\leq i\leq b$）。本文证明对于任意$q\geq 3$及$H(n,q)$中任意$b=\lfloor(1-\frac1q)n\rfloor$个顶点的序列$v_1,\ldots,v_b$，同样存在一个顶点，其到$v_i$的距离至少为$b-i+1$（对所有$1\leq i\leq b$）。阿隆的证明使用了贝克-斯宾塞引理，该引理基于贝克和菲亚拉在研究组合差异时引入的浮动变量法。为完成我们的证明，我们借助多尔和斯里瓦斯塔夫提出的多色浮动变量法，对贝克-斯宾塞引理进行了推广。