Improved bounds for the sunflower lemma

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 of them. Erdős and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w w sets, must contain a sunflower with r petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to c w for some constant c. In this paper, we improve the bound to about (log w) w . In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is sharp up to lower order terms.