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

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.