Erdős–Ko–Rado theorem for {0,±1}-vectors

The main object of this paper is to determine the maximum number of {0,±1}-vectors subject to the following condition. All vectors have length n, exactly k of the coordinates are +1 and one is −1, n≥2k. Moreover, there are no two vectors whose scalar product equals the possible minimum, −2. Thus, th...

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Bibliographic Details
Published inJournal of combinatorial theory. Series A Vol. 155; pp. 157 - 179
Main Authors Frankl, Peter, Kupavskii, Andrey
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2018
Subjects
[formula omitted]-vectors
Antipodal vectors
Erdos–Ko–Rado theorem
Intersecting family
{0,±1}-vectors
Erdos–Ko–Rado theorem
Antipodal vectors
Intersecting family
Online AccessGet full text
ISSN0097-3165
1096-0899
DOI10.1016/j.jcta.2017.11.003

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Summary:The main object of this paper is to determine the maximum number of {0,±1}-vectors subject to the following condition. All vectors have length n, exactly k of the coordinates are +1 and one is −1, n≥2k. Moreover, there are no two vectors whose scalar product equals the possible minimum, −2. Thus, this problem may be seen as an extension of the classical Erdős–Ko–Rado theorem. Rather surprisingly there is a phase transition in the behaviour of the maximum at n=k2. Nevertheless, our solution is complete. The main tools are from extremal set theory and some of them might be of independent interest.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2017.11.003