Here is a problem I'm trying to solve:
Let $G$ be a graph on at least $2k+2$ vertices which has a perfect matching. Show that if every set of $k$ independent edges is included in a perfect matching then every set of $k−1$ independent edges is included in a perfect matching.
Here is my attempt: By tutte berge theorem: $comp(G\ X) ≤|X|+|V(G)|−2k$ if and only if $G$ has a matching of size $k$.
Ok So I assume a perfect matching in $G$ has size at least $k$. Let $S$ be a set of independent edges of size $k-1$, suppose it is not in a set of independent edges of size $k$, then every other edges in $G$ is incident to an end vertices of edges in $S$. But then $S$ could not be inside of a perfect matching of size at least $k$? What is wrong here?
Help, you can also give a complete solution.