Improved bounds for the sunflower lemma

Abstract

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. ErdA'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.