Rainbow Matchings in Properly Edge-Coloured Multigraphs

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n+1 edges of each colour there must be a matching that uses each colour exactly once (such a matching is called rainbow). This conjecture recently have been proved asymptotically by Pokrovskiy. In this talk, I will consider the same question without the bipartiteness assumption. It turns out that in any multigraph with bounded edge multiplicities, that is properly edge-coloured by n colours with at least n+o(n) edges of each colour, there must be a matching of size n-O(1) that uses each colour at most once. This is joint work with Peter Keevash.