向日葵引理的改进界|数学年鉴

IF 5.7 1区数学 Q1 MATHEMATICS

Annals of Mathematics Pub Date : 2021-11-02 DOI:10.4007/annals.2021.194.3.5

Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang

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引用次数: 0

摘要

一朵有$r$花瓣的向日葵是$r$组花瓣的集合，因此每一对花瓣的交点等于所有花瓣的交点。Erdős和Rado证明了向日葵引理:对于任何固定的$r$，任何大小为$w$的集合族，至少有$w^w$的集合，必须包含有$r$花瓣的向日葵。著名的向日葵猜想指出，对于某个常数c，集合数的界可以改进为c^w。在本文中，我们改进了这个界约$(\log\， w)^w$。事实上，我们证明了一个鲁棒的向日葵概念的结果，对于这个概念，我们得到的界在低阶项上是尖锐的。

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Improved bounds for the sunflower lemma | Annals of Mathematics

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.

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来源期刊

Annals of Mathematics 数学-数学

CiteScore

9.10

自引率

2.00%

发文量

审稿时长

12 months

期刊介绍： The Annals of Mathematics is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Founded in 1884 by Ormond Stone of the University of Virginia, the journal was transferred in 1899 to Harvard University, and in 1911 to Princeton University. Since 1933, the Annals has been edited jointly by Princeton University and the Institute for Advanced Study.