Even after the breakthrough work of Croot-Lev-Pach and Ellenberg-Gijswijt, the size of the largest subsets of F_p^n without 3-term arithmetic progressions is not known. I will explain how, with Robert Kleinberg and David Speyer, we were able to find essentially the largest possible subsets with a relaxed version of that property. I will also discuss the relationship to fast matrix multiplication algorithms.