Improved bounds for the sunflower lemma

TL;DR

This paper improves the upper bound on the number of sets needed to guarantee a sunflower with r petals, reducing it from exponential to a polylogarithmic function of the set size, and proves the bound is nearly tight.

Contribution

It provides a significantly improved bound for the sunflower lemma, approaching the conjectured optimal bound, for a robust sunflower notion.

Findings

Bound improved to about (log w)^w

Result is sharp up to lower order terms

Applicable to a robust sunflower concept

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. 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 $w^w$ 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.