30 research outputs found
On Sunflowers and Matrix Multiplication
We present several variants of the sunflower conjecture of Erdős and Rado and discuss the relations among them.
We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. 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 and Winograd; we also formulate a “multicolored” sunflower conjecture in Zn₃ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (although it does not seem to impact a second conjecture in that paper 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 Zn₃ is a strengthening of the well-known (ordinary) sunflower conjecture in Zn₃, and we show via our connection that a construction of Cohn et al. 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 on the size of the largest 3-sunflower-free set in Zn₃
Upper bounds for sunflower-free sets
A collection of sets is said to form a -sunflower, or -system,
if the intersection of any two sets from the collection is the same, and we
call a family of sets sunflower-free if it contains no
sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and
Croot, Lev and Pach we apply the polynomial method directly to
Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free
family of subsets of has size at most We say that
a set for is
sunflower-free if every distinct triple there exists a coordinate
where exactly two of are equal. Using a version of the
polynomial method with characters
instead of polynomials, we
show that any sunflower-free set has size
where . This can be
seen as making further progress on a possible approach to proving the
Erd\H{o}s-Rado sunflower conjecture, which by the work of Alon, Sphilka and
Umans is equivalent to proving that for some constant
independent of .Comment: 5 page