On generalized corners and matrix multiplication
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Suppose that S ⊆ [n]^2 contains no three points of the form (x,y), (x,y+δ), (x+δ,y'), where δ≠ 0. How big can S be? Trivially, n ≤ |S| ≤ n^2. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that |S| ≤ O(n^2/(loglog n)^c) for some small c > 0, and a construction due to Petrov [Pet23], which shows that |S| ≥Ω(n log n/√(loglog n)). Could it be that for all ε > 0, |S| ≤ O(n^1+ε)? We show that if so, this would rule out obtaining ω = 2 using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on ω to date), for which no barriers are currently known. Furthermore, an upper bound of O(n^4/3 - ε) for any fixed ε > 0 would rule out a conjectured approach to obtain ω = 2 of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.
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