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arXiv · 1908.08483

Improved bounds for the sunflower lemma

Ryan Alweiss·Shachar Lovett·Kewen Wu·Jiapeng Zhang
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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.

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Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang. 2019-08-22. Improved bounds for the sunflower lemma. https://arxiv.org/abs/1908.08483

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