  Aharoni and Berger conjectured that in every proper edge-colouring of a
bipartite multigraph by $n$ colours with at least $n+1$ edges of each colour
there is a rainbow matching using every colour. This conjecture generalizes a
longstanding problem of Brualdi and Stein about transversals in Latin squares.
Here an approximate version of the Aharoni-Berger Conjecture is proved---it is
shown that if there are at least $n+o(n)$ edges of each colour in a proper
$n$-edge-colouring of a bipartite multigraph then there is a rainbow matching
using every colour.
