Summary: On Sunflowers and Matrix Multiplication
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 ques-
tions of Coppersmith and Winograd [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 Winograd [CW90]; we also
formulate a "multicolored" sunflower conjecture in Zn
3 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-Winograd
conjecture actually implies the Cohn et al. conjecture.
The multicolored sunflower conjecture in Zn
3 is a strengthening of the well-known