  Many well-known problems in Combinatorics can be reduced to finding a large
rainbow structure in a certain edge-coloured multigraph. Two celebrated
examples of this are Ringel's tree packing conjecture and Ryser's conjecture on
transversals in Latin squares. In this paper, we answer such a question raised
by Grinblat twenty years ago. Let an $(n,v)$-multigraph be an $n$-edge-coloured
multigraph in which the edges of each colour span a disjoint union of
non-trivial cliques that have in total at least $v$ vertices. Grinblat
conjectured that for all $n \geq 4$, every $(n,3n-2)$-multigraph contains a
rainbow matching of size $n$. Here, we prove the conjecture for all
sufficiently large $n$.
