Abstract 
A subset $A$ of a metric space $(X,d)$ is called thick if, for every
$r>0$, there is $a\in A$ such that $B_{d}(a,r)\subseteq A,$ where
$B_{d}(a,r)=\{x\in X\colon d(x,a)\leq r\}$. We show that if $(X, d)$ is
unbounded and has no asymptotically isolated balls then, for each
$r>0$, there exists a partition $X=X_{1}\cup X_{2}$ such that
$B_{d}(X_{1},r)$ and $B_{d}(X_{2},r)$ are not thick. 
Reference 
1. T. Banakh, I.V. Protasov, S. Slobodianiuk, Subamenable groups and their partitions, preprint (http://
arxiv.org/abs/1210.5804).
2. T. Banakh, I. Zarichnyi, The coarse characterization of homogeneous ultrametric space, Groups, Geometry
and Dynamics, 5 (2011), 691–728.
3. I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser.,
V.11, VNTL Publisher, Lviv, 2003.
4. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser. V.12, VNTL Publisher, Lviv,
2007.
