  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.
