In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.

A collection of k sets is said to form a k-sunflower, or ∆-system, if the intersection of any two sets from the collection is the same, and we call a family of sets F sunflower-free if it contains no 3-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt ('On large subsets of F

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Let A ⊂ {1, . . . , N } be a set of prime numbers containing no nontrivial arithmetic progressions. Suppose that A has relative density α = |A|/π(N ), where π(N ) denotes the number of primes in the set {1, . . . , N }. By modifying Helfgott and De Roton's work [Improving Roth's theorem in the primes. Int. Math. Res. Not. IMRN 2011 (4) (2011), 767-783], we improve their bound and show thatIn 1939, Van Der Corput [13] showed that P contains infinitely many nontrivial three-term arithmetic progressions. Green [4] proved an analogue of