Strong bounds for skew corner-free sets

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of [n] × [n] could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with h ≠ 0. We prove any skew corner-free set has size at most exp(-Ω(log1/12 n))· n², nearly matching the best known lower bound of exp(-O(√log n))· n² by Beker (arXiv'24). Our techniques generalize those of Kelley and Meka's recent breakthrough on three-term arithmetic progression (FOCS'23), answering a question of Beker (arXiv'24). We note that a similar bound was obtained concurrently and independently by Mili\'cevi\'c (arXiv'24).