Good bounds for sets lacking skew corners

A skew corner is a triple of points in Z × Z of the form $(x,y), (x, y + a)$ and $(x + a, y')$. Pratt posed the following question: how large can a set A ⊆ [n] × [n] be, provided it contains no non-trivial skew corner (i.e. one for which a=0)? We prove that |A| ≤ exp(- clogᶜ n) n², for an absolute constant $c > 0$, which, along with a construction of Beker, essentially resolves Pratt's question. Our argument is represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth's theorem.