On the arithmetic mean of the size of cross-union families
type: Item
instance of: scholarly article
title: On the arithmetic mean of the size of cross-union families
zbMATH Open document ID: 1488.05478
published in: Acta Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: The celebrated Erdős-Ko-Rado theorem provides an upper bound on the size of an intersecting family; its generalization by \textit{L. Pyber} [J. Comb. Theory, Ser. A 43, 85--90 (1986; Zbl 0614.05001)] deals with the size of cross-intersecting families.\par The families \(\mathcal{F}_{0},\mathcal{F}_{1},\dots,\mathcal{F}_{s}\) of \(k\)-subsets of \([n]\) are cross-union if \(F_{0}\cup F_{1}\cup \dots\cup F_{s}\neq \lbrack n]\) for all choices of \(F_{i}\in \mathcal{F}_{i}.\) \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)] proved that for \(n\leq k(s+1)\) the geometric mean of \(\left\vert \mathcal{F}_{i}\right\vert \) is at most \(\binom{n-1}{k}\). It is conjectured that the same is true for the arithmetic mean for the range \(ks<n<k(s+1)\), \(s>s_{0}(k)\). As a support, the conjecture is proved in several cases.
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 7377542
zbMATH Keywords: finite set
zbMATH Keywords: intersection
zbMATH Keywords: cyclic permutation
author: Peter Frankl
reviewed by: Peter Horák
MaRDI profile type: Publication
full work available at URL: s10474-021-01138-6
OpenAlex ID: W3159734514
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DOI: 10.1007/S10474-021-01138-6
description: scientific article; zbMATH DE number 7377542