  Assuming $V=L$, we construct a plane set $E$ of Hausdorff dimension $1$ whose
every orthogonal projection onto straight lines through the origin has
Hausdorff dimension $0$. This is a counterexample to J. M. Marstrand's seminal
projection theorem. While counterexamples had already been constructed decades
ago, initially by R. O. Davies, the novelty of our result lies in the fact that
$E$ is co-analytic. Following Marstrand's original proof (and R. Kaufman's
newer, and now standard, approach based on capacities), a counterexample to the
projection theorem cannot be analytic, hence our counterexample is optimal. We
then extend the result in a strong way: we show that for each $\epsilon \in (0,
1)$ there exists a co-analytic set $E_{\epsilon}$ of dimension $1 + \epsilon$,
each of whose orthogonal projections onto straight lines through the origin has
Hausdorff dimension $\epsilon$. The constructions of $E$ and $E_{\epsilon}$ are
by induction on the countable ordinals, applying a theorem by Z. Vidnyanszky.
