  A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point
of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body $C$, there exists a constant $m=m(C)$ such that every
$m$-fold covering of the plane with translates of $C$ splits into $2$
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every $m$, an
unsplittable $m$-fold covering of the plane with translates of any open convex
body $C$ which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave,
they satisfy the conjecture: every $3$-fold covering of any region of the plane
by translates of such a set $C$ splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant $c>0$ such that, for
any positive integer $m$, every $m$-fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} $c2^{m/2}$ sets.
