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 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.

Took into account comments from the Annals of Mathematics

References in corpus (2)

• Note on Sunflowers
• Improved Bound on Sets Including No Sunflower with Three Petals

Cited by in corpus (2)

• Coding for Sunflowers
• The Karger-Stein Algorithm is Optimal for $k$-cut