  Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most $k$ sets. Suppose there is a function
$f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine
at most $f(n)$ holes, which means that the complement of their union has at
most $f(n)$ bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that $\mathcal{X}$ can be
decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant
depending only on $k$ and $f$.
