  Let $\mathcal C$ be the Cantor set. For each $n\geqslant 3$ we construct an
embedding $A: \mathcal C \times \mathcal C \to \mathbb R^n$ such that
$A(\mathcal C \times \{s\})$, for $s\in\mathcal C$, are pairwise ambiently
incomparable everywhere wild Cantor sets (generalized Antoine's necklaces).
This serves as a base for another new result proved in this paper: for each
$n\geqslant 3$ and any non-empty perfect compact set $X$ which is embeddable in
$\mathbb R^{n-1}$, we describe an embedding $\mathbb A : X \times \mathcal C
\to \mathbb R^n$ such that each $\mathbb A (X \times \mathcal \{s\} )$, $s\in
\mathcal C$, contains the corresponding $A (\mathcal C \times \{s\} )$, and is
``nice'' on the complement $\mathbb A (X \times \mathcal \{s\} )-A (\mathcal C
\times \{s\} )$; in particular, the images $\mathbb A ( X \times \{s\})$, for
$s\in\mathcal C$, are ambiently incomparable pairwise disjoint copies of $X$.
This generalizes and strengthens theorems of J.R.Stallings (1960), R.B.Sher
(1968), and B.L.Brechner-J.C.Mayer (1988).
