  We construct skew corner-free subsets of $[n]^2$ of size
$n^2\exp(-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form
$\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. In the other direction, we
prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some
absolute constant $c > 0$. This improves on the previously best known upper
bound, coming from Shkredov's work on the corners theorem.
