@inproceedings{85b4950b74ed4862af5620e5ea83557b,

title = "On sunflowers and matrix multiplication",

abstract = "We present several variants of the sunflower conjecture of Erdos and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Wino grad [CW90] and Cohn et al [CKSU05] regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erdos-Rado sunflower conjecture (if true) implies a negative answer to the ''no three disjoint equivoluminous subsets'' question of Coppersmith and Wino grad [CW90]; we also formulate a ''multicolored'' sunflower conjecture in Z-3 n and show that (if true) it implies a negative answer to the ''strong USP'' conjecture of [CKSU05] (although it does not seem to impact a second conjecture in [CKSU05] or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith-Wino grad conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in Z-3 n is a strengthening of the well-known (ordinary) sunflower conjecture in Z-3 n, and we show via our connection that a construction from [CKSU05] yields a lower bound of (2.51\ldots) n on the size of the largest {\em multicolored} 3-sunflower-free set, which beats the current best known lower bound of (2.21\ldots) n [Edel04] on the size of the largest 3-sunflower-free set in Z-3 n.",

keywords = "Matrix Multiplication, Sunflower Conjecture",

author = "Noga Alon and Amir Shpilka and Christopher Umans",

year = "2012",

doi = "10.1109/CCC.2012.26",

language = "English (US)",

isbn = "9780769547084",

series = "Proceedings of the Annual IEEE Conference on Computational Complexity",

pages = "214--223",

booktitle = "Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012",

note = "IEEE Computer Society Technical Committee on Mathematical Foundations of Computing ; Conference date: 26-06-2012 Through 29-06-2012",

}