We show that for every
s
∈
[
n
−
2
,
n
]
s \in [n - 2,n]
there exists a homogeneously embedded wild Cantor set
C
s
{C^s}
in
R
n
,
n
≥
3
\mathbb {R}^n, n \geq 3
, of (local) Hausdorff dimension
s
s
. Also, it is shown that for every
s
∈
[
n
−
2
,
n
]
s \in [n - 2,n]
and for any integer
k
≠
n
k \ne n
such that
1
≤
k
≤
s
1 \leq k \leq s
, there exist everywhere wild
k
k
-spheres and
k
k
-cells, in
R
n
,
n
≥
3
\mathbb {R}^n, n \geq 3
, of (local) Hausdorff dimension
s
s
.