A generalization of the Katona theorem for cross t-intersecting families
type: Item
instance of: scholarly article
title: A generalization of the Katona theorem for cross t-intersecting families
zbMATH Open document ID: 0672.05001
DOI: 10.1007/BF01788667
author: Makoto Matsumoto
author: Norihide Tokushige
published in: Graphs and Combinatorics
publication date: 1989-01-01T00:00:00Z
review text: Let X be an n-element set and let k,t\(\leq n\) be integers. Write \(K^ n_ k\) for the number of at least k-size subsets of X. Let \({\mathcal A}\) and \({\mathcal B}\) be families of subsets of X such that \(| A\cap B| \geq t\) for every A in \({\mathcal A}\) and B in \({\mathcal B}\). In this paper it is shown that (i) If \(n+t=2k\) then \(| {\mathcal A}| | {\mathcal B}| \leq (K^ n_ k)^ 2,\) (ii) If \(n+t=2k-1\) then \[ | {\mathcal A}| | {\mathcal B}| \leq \max \{(K^ n_ k+\left( \begin{matrix} n-1\\ k- 1\end{matrix} \right))^ 2,\quad K^ n_ k K^ n_{k-1}\}. \] These results generalize similar results for the case when \({\mathcal A}={\mathcal B}\) of \textit{G. O. H. Katona} [Acta Math. Acad. Sci. Hung. 15, 329-337 (1964; Zbl 0134.251)].
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 4099300
zbMATH Keywords: n-element set
zbMATH Keywords: k-size subsets
reviewed by: Neil H. Williams
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01788667
OpenAlex ID: W2024552814
description: scientific article; zbMATH DE number 4099300