  Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb
R}^d$. For any $t \in [0,1]$ we construct a compact set $\Gamma\subset{\mathbb
R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each
$\ell_i$, such that the image of each projection has dimension $t$. This
immediately implies the existence of homeomorphisms between certain Cantor-type
sets whose graphs have large dimensions. As an application, we construct a
collection $E$ of disjoint, non-parallel $k$-planes in $\mathbb{R}^d$, for $d
\geq k+2$, whose union is a small subset of $\mathbb{R}^d$, either in Hausdorff
dimension or Lebesgue measure, while $E$ itself has large dimension. As a
second application, for any countable collection of vertical lines $w_i$ in the
plane we construct a collection of nonvertical lines $H$, so that $F$, the
union of lines in $H$, has positive Lebesgue measure, but each point of each
line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff
dimension of $F\cap w_i$ is zero.
