This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ where each $U^i$ consists of uniformly bounded $\lambda$-disjoint sets $U_j^i$ ("$\lambda$-disjoint" means $dist(U_j^i, U_l^i)\ge \lambda$ for every $i,j\not=l$), $\lambda$ is fixed, $>1$?

In http://front.math.ucdavis.edu/1008.3868, we show (Lemma 3.7) that for $k<2^{\lambda-1}$, the minimal $c$ is $k$. The question is what happens for $k\ge 2^{\lambda-1}$.

Update: The answer is not known even if $\lambda=3$.