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

- 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

- Bollobás B, Daykin DE, Erdős P (1976) Sets of independent edges of a hypergraph. Q J Math Oxf Ser (2) 27:25–32MathSciNetCrossRefGoogle Scholar
- Erdős P (1965) A problem on independent r-tuples. Ann Univ Sci Bp Eotvos Sect Math 8:93–95MathSciNetzbMATHGoogle Scholar
- Erdős P, Gallai T (1959) On maximal paths and circuits of graphs. Acta Math Acad Sci Hung 10:337–56MathSciNetCrossRefGoogle Scholar
- Erdős P, Ko C, Rado R (1961) Intersection theorems for systems of finite sets. Q J Math Oxf Ser 2(12):313–318MathSciNetCrossRefGoogle Scholar
- Frankl P (2013) Improved bounds for Erdős’ matching conjecture. J Combin Theory Ser A 120:1068–1072MathSciNetCrossRefGoogle Scholar
- Frankl P (2017) On the maximum number of edges in a hypergraph with given matching number. Discrete Appl Math 216:562–581MathSciNetCrossRefGoogle Scholar
- Frankl P, Łuczak T, Mieczkowska K (2012a) On matchings in hypergraphs. Electron J Combin 19(2):42MathSciNetzbMATHGoogle Scholar
- Frankl P, Rődl V, Ruciński A (2012b) On the maximum number of edges in a triple system not containing a disjoint family of a given size. Combin Probab Comput 21:141–148MathSciNetCrossRefGoogle Scholar
- Frankl P, Rődl V, Ruciński A (2017) A short proof of Erdős’ conjecture for triple systems. Acta Math Hung 151:495–509CrossRefGoogle Scholar
- Huang H, Loh P, Sudakov B (2012) The size of a hypergraph and its matching number Combin. Probab Comput 21:442–450MathSciNetCrossRefGoogle Scholar
- Ku CY, Leader I (2006) An Erdös–Ko–Rado theorem for partial permutations. Discrete Math 306:74–86MathSciNetCrossRefGoogle Scholar
- Li YS, Wang J (2007) Erdős–Ko–Rado-type theorems for colored sets. Elecron J Combin 14:R1zbMATHGoogle Scholar
- Łuczak T, Mieczkowska K (2014) On Erdős’ extremal problem on matchings in hypergraphs. J Combin Theory Ser A 124:178–194MathSciNetCrossRefGoogle Scholar