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

A labeled set is a set of distinct elements with labels assigned to the elements. The family $${{\mathcal {L}}}^k_{n,p}$$ L n , p k 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 $$1\le s\le p-1$$ 1 ≤ s ≤ p - 1 , if a family $${\mathcal {F}}\subseteq {\mathcal {L}}^k_{n,p}$$ F ⊆ L n , p k does not have more than s pairwise disjoint members, then $$|{\mathcal {F}}|\le sp^{k-1}\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) $$ | F | ≤ s p k - 1 n - 1 k - 1 . Furthermore, equality holds if and only if $${\mathcal {F}}$$ F has the form $${{\mathcal {F}}}_{i,S}$$ F i , S , where $${{\mathcal {F}}}_{i,S}$$ F i , S is the family of all members of $${{\mathcal {L}}}^k_{n,p}$$ L n , p k containing the element i and using labels only within the s-set S.