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Intersecting families without unique shadow

Part of: Extremal combinatorics

Published online by Cambridge University Press:  02 October 2023

Peter Frankl  and
Jian Wang
Peter Frankl
Affiliation:
Rényi Institute, Budapest, Hungary
Jian Wang*
Affiliation:
Department of Mathematics, Taiyuan University of Technology, Taiyuan, P. R. China
*
Corresponding author: Jian Wang; Email: wangjian01@tyut.edu.cn
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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.

Keywords

Finite setsintersectionshadow

MSC classification

Primary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 33 , Issue 1 , January 2024 , pp. 91 - 109
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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