Restricted \(t\)-wise \(\mathcal{L}\)-intersecting families on set systems
type: Item
instance of: scholarly article
title: Restricted \(t\)-wise \(\mathcal{L}\)-intersecting families on set systems
zbMATH Open document ID: 1234.05042
author: Jiuqiang Liu
author: Larry X. W. Wang
author: Qingde Kang
published in: European Journal of Combinatorics
publication date: 2011-03-04T00:00:00Z
review text: Let \(L\) be any ordered set of \(s\) nonnegative integers, \(\lambda=\min L\) and \(t\geq2\). A family \(F\) of subsets of \([n]\) is \textit{\(t\)-wise \(L\)-intersecting} if the size of the intersection of any \(t\) distinct members of \(F\) belongs to \(L\). The authors show that {\parindent=5mm \begin{itemize}\item[1)]if \(t\geq 3\) and \(F\) is a \(t\)-wise \(L\)-intersecting family of subsets from \([n]\) then either \[ \left|\bigcap_{A\in F}A\right|< \lambda\text{ and }|F|=O(n^s) \] or \[ \left|\bigcap_{A\in F}A\right|\geq\lambda\text{ and }|F|\leq \frac{k+s-1}{s+1}\binom{n-\lambda}{s}+\sum_{j=0}^{s-1}\binom{n-\lambda}{j} \] \item[2)]if \(F\) is a \(k\)-uniform \(t\)-wise \(\{\lambda\}\)-intersecting family of subsets from \([n]\) then \(|F|\leq\frac{(n-\lambda)(t-1)}{k-\lambda}\) for all \(n>\lambda+k+\frac{k(k-2)}{t-1}\). \end{itemize}}
reviewed by: Toufik Mansour
Mathematics Subject Classification ID: 05A20
zbMATH DE Number: 5860467
zbMATH Keywords: intersection family
zbMATH Keywords: \(k\)-wise \(L\)-intersecting family
MaRDI profile type: Publication
full work available at URL: j.ejc.2010.11.009
OpenAlex ID: W1982737120
cites work: A Note on Fisher's Inequality for Balanced Incomplete Block Designs
cites work: Intersection Properties of Systems of Finite Sets
cites work: Intersection theorems with geometric consequences
cites work: Extremal set systems with restricted \(k\)-wise intersections.
cites work: On \(k\)-wise set-intersections and \(k\)-wise Hamming-distances
cites work: On t-designs
DOI: 10.1016/J.EJC.2010.11.009
description: scientific article; zbMATH DE number 5860467