On skew corner-free sets

We construct skew corner-free sets in $[n{]}^2$ of size $n^{5\mathrm{/}4}$, thereby disproving a conjecture of Kevin Pratt. We also show that any skew corner-free set in ${\mathbb{F}}_q^n\times {\mathbb{F}}_q^n$ must have size at most $q^{(2-c)n}$, for some positive constant $c$ which depends on $q$.