An improved upper bound for the size of a sunflower-free family
An improved upper bound for the size of a sunflower-free family
复制标题
无向日葵家庭规模的改进上限
DOI：
--
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
G. Hegedus
中科院分区：
文献类型：
--
作者：
G. Hegedus
这里我们结合了陶渊明的条秩界方法和Gröbner基技巧，并将其应用于ERDőS-Rado向日葵猜想。让0≤k≤n\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\Usepackage{amsFonts}\Usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setLong{\oddsidemargin}{-69pt}\Begin{Document}$${leq k\leq n}$$\end{Document}为整数。我们证明了如果F\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\begin{document}$${\mathcal{F}}$$\end{document}是[n]的k-一致子集族，不含3瓣向日葵，然后|F|≤3n⌊n/3⌋.\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$|\Mathcal{F}|\leq3\Leq3\Leq3\rFloor\END{ARRAY}\)。$$\end{文档}这一结果允许我们略微改进了最近Naslund和Sawin关于2[n]中无葵花族的大小的一个上界。
We combine here Tao’s slice-rank bounding method and Gröbner basis techniques and apply it to the Erdős–Rado Sunflower Conjecture. Let 0≤k≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0\leq k\leq n}$$\end{document} be integers. We prove that if F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document} is a k-uniform family of subsets of [n] without a sunflower with 3 petals, then |F|≤3n⌊n/3⌋.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal{F}|\leq3 \left(\begin{array}{c} {n }\\ \lfloor n/3\rfloor \end{array}\right).$$\end{document}This result allows us to improve slightly a recent upper bound of Naslund and Sawin for the size of a sunflower-free family in 2[n].