On sunflowers and matrix multiplication

Abstract

We present several variants of the sunflower conjecture of Erdős & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erdős–Rado sunflower conjecture (if true) implies a negative answer to the "no three disjoint equivoluminous subsets" question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a "multicolored" sunflower conjecture in Z^n_3 and show that (if true) it implies a negative answer to the "strong USP" conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in Z^n_3 is a strengthening of the well-known (ordinary) sunflower conjecture in Z^n_3 , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51...)^n on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21...)^n Edel (2004) on the size of the largest 3-sunflower-free set in Z^n_3.

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