### 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.

Original language | English (US) |
---|---|

Title of host publication | Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012 |

Pages | 214-223 |

Number of pages | 10 |

DOIs | |

State | Published - 2012 |

Event | IEEE Computer Society Technical Committee on Mathematical Foundations of Computing - Porto, Portugal Duration: Jun 26 2012 → Jun 29 2012 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
---|---|

ISSN (Print) | 1093-0159 |

### Other

Other | IEEE Computer Society Technical Committee on Mathematical Foundations of Computing |
---|---|

Country | Portugal |

City | Porto |

Period | 6/26/12 → 6/29/12 |

### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

### Keywords

- Matrix Multiplication
- Sunflower Conjecture

## Fingerprint Dive into the research topics of 'On sunflowers and matrix multiplication'. Together they form a unique fingerprint.

## Cite this

*Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012*(pp. 214-223). [6243397] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2012.26