  We study the possibility of realizing exotic smooth structures on punctured
simply connected $4$-manifolds as leaves of a codimension one foliation on a
compact manifold. In particular, we show the existence of uncountably many
smooth open $4$-manifolds which are not diffeomorphic to any leaf of a
codimension one transversely $C^{2}$ foliation on a compact manifold. These
examples include some exotic ${\mathbb R}^4$'s and exotic cylinders
$S^3\times{\mathbb R}$.
