Intersecting families without unique shadow

IF 0.9 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Combinatorics, Probability & Computing Pub Date : 2023-10-02 DOI:10.1017/s0963548323000305

Peter Frankl, Jian Wang

引用次数: 1

Abstract

Abstract Let $\mathcal{F}$ be an intersecting family. A $(k-1)$ -set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$ . Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$ . In the present paper, we show that for $n\geq 28k$ , $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.

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交叉的家族没有独特的影子

让$\mathcal{F}$成为一个交叉的家庭。如果一个$(k-1)$ -set $E$恰好包含在$\mathcal{F}$的一个成员中，则它被称为唯一影子。让${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$。在本文中，我们证明了对于$n\geq 28k$, $\mathcal{A}$是在没有唯一阴影的所有相交族中达到最大大小的唯一族。其他几个类似味道的结果也得到了证实。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Combinatorics, Probability & Computing 数学-计算机：理论方法

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.

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