A sunflower with r petals is a collection of r sets so that the
intersection of each pair is equal to the intersection of all. Erd\H{o}s and
Rado proved the sunflower lemma: for any fixed r, any family of sets of size
w, with at least about ww sets, must contain a sunflower. The famous
sunflower conjecture is that the bound on the number of sets can be improved to
cw for some constant c. In this paper, we improve the bound to about
(logw)w. In fact, we prove the result for a robust notion of sunflowers,
for which the bound we obtain is tight up to lower order terms.Comment: Revised preprint, added sections on applications and rainbow
sunflower