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.


Ryan Alweiss

Department of Mathematics, Princeton University, Princeton, NJ

Shachar Lovett

Department of Computer Science & Engineering, University of California, San Diego, La Jolla, CA

Kewen Wu

Department of Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA

Jiapeng Zhang

Department of Computer Science, University of Southern California, Los Angeles, CA