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Journal of Combinatorial Optimization

, Volume 38, Issue 4, pp 1296–1304 | Cite as

On the maximum size of subfamilies of labeled set with given matching number

  • Huajun ZhangEmail author
Article
  • 22 Downloads

Abstract

A labeled set is a set of distinct elements with labels assigned to the elements. The family \({{\mathcal {L}}}^k_{n,p}\) is the family of labeled k-element subsets of [n] with labels chosen from a set of size p. Two labeled sets are disjoint if they do not share an element that has the same label in both sets. For \(1\le s\le p-1\), if a family \({\mathcal {F}}\subseteq {\mathcal {L}}^k_{n,p}\) does not have more than s pairwise disjoint members, then \(|{\mathcal {F}}|\le sp^{k-1}\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \). Furthermore, equality holds if and only if \({\mathcal {F}}\) has the form \({{\mathcal {F}}}_{i,S}\), where \({{\mathcal {F}}}_{i,S}\) is the family of all members of \({{\mathcal {L}}}^k_{n,p}\) containing the element i and using labels only within the s-set S.

Keywords

Intersecting family Erdős–Ko–Rado theorem Matching number 

Mathematics Subject Classification

05D05 06A07 

Notes

References

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China

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