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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改进了向日葵引理的边界

具有$r$花瓣的向日葵是$r$集合的集合，因此每对的交集等于所有的交集。Erd\H{o}s 和 Rado 证明了向日葵引理：对于任何固定的 $r$，任何一组大小为 $w$ 的集合，至少有大约 $w^w$ 的集合，必须包含一个向日葵。著名的向日葵猜想是对于某些常数 $c$，集合数的界限可以改进为 $c^w$。在本文中，我们将边界改进为大约 $(\log w)^w$。事实上，我们证明了向日葵的稳健概念的结果，我们获得的界限在低阶项上是紧的。