arXiv - CS - Computational Complexity Pub Date : 2019-08-22 , DOI: arxiv-1908.08483
Ryan Alweiss; Shachar Lovett; Kewen Wu; Jiapeng Zhang

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. 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. The famous sunflower conjecture is 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 tight up to lower order terms.

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