  Erd\H{o}s and Hanani proved that for every fixed integer $k \ge 2$, the
complete graph $K_n$ can be almost completely packed with copies of $K_k$; that
is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an
$o_n(1)$ fraction of its edges. Equivalently, elements of the set $\C(k)$ of
all red-blue edge colorings of $K_k$ can be used to almost completely pack
every red-blue edge coloring of $K_n$.
  The following strengthening of the aforementioned Erd\H{o}s-Hanani result is
considered. Suppose $\C' \subset \C(k)$. Is it true that we can use elements
only from $\C'$ and almost completely pack every red-blue edge coloring of
$K_n$? An element $C \in \C(k)$ is {\em avoidable} if $\C'=\C(k) \setminus C$
has this property and a subset ${\cal F} \subset \C(k)$ is avoidable if
$\C'=\C(k) \setminus {\cal F}$ has this property.
  It seems difficult to determine all avoidable graphs as well as all avoidable
families. We prove some nontrivial sufficient conditions for avoidability. Our
proofs imply, in particular, that (i) almost all elements of $\C(k)$ are
avoidable (ii) all Eulerian elements of $\C(k)$ are avoidable and, in fact, the
set of all Eulerian elements of $\C(k)$ is avoidable.
