For given $n>1$ I'm looking for the maximum number $k$ of colors such that there exists a $k$-coloring of the edges of the complete graph on $2n$ vertices with the property that a copy of every possible $k$-colored perfect matching is present in the colored complete graph.
For example, we know that there are $(n+1)$ 2-colored perfect matchings. On the other hand, we know that the complete graph can be decomposed into $(2n-1)$ perfect matchings (cf. Theorem 3.2 in 1-factorizations by Wallis). This gives $k>1$. On the other hand, for $(2n-1)$ colors we need to give each of the disjoint perfect matchings a dedicated color to accomodate the monochromatic perfect matchings. But then a perfect matching with $(n-1)$ edges of color $1$ and one edge of color $2$ cannot be present. So we have $k<2n-1$.
Can we improve upon these bounds?
EDIT: The problem seems to be harder than I thought, so I narrow it down.
Prove or disprove that $k=\omega(1)$.