Minki Kim (김민기), Rainbow paths and rainbow matchings

We prove that if $n \geq 3$, then any family of $3n-3$ sets of matchings of size $n$ in any graph has a rainbow matching of size $n$. This improves on a previous result, in which $3n-3$ is replaced by $3n-2$. We also prove a “cooperative” generalization: for $t > 0$ and $n \geq 3$, any $3n-4+t$ sets of edges, the union of every $t$ of which contains a matching of size $n$, have rainbow matching of size $n$. This is joint work with Ron Aharoni, Joseph Briggs, and Jinha Kim.