  We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty
collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that
every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least
$\alpha$ with $k-1 < \alpha \leq k$, then $\dim B \geq \alpha +\dim E/(k+1)$,
where $\dim$ denotes the Hausdorff dimension. This estimate generalizes the
well known Furstenberg-type estimate that every $\alpha$-Furstenberg set in the
plane has Hausdorff dimension at least $\alpha + 1/2$.
  More generally, we prove that if $B$ and $E$ are as above with $0 < \alpha
\leq k$, then $\dim B \geq \alpha +(\dim E-(k-\lceil \alpha
\rceil)(n-k))/(\lceil \alpha \rceil+1)$. We also show that this bound is sharp
for some parameters.
  As a consequence, we prove that for any $1 \leq k<n$, the union of any
nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces
of $\mathbb{R}^n$ has Hausdorff dimension at least $k+\frac{s}{k+1}$.
