Large matchings in bipartite graphs have a rainbow matching

Let g ( n ) be the least number such that every collection of n matchings, each of size at least g ( n ) , in a bipartite graph, has a full rainbow matching. Aharoni and Berger (2009) conjectured that g ( n ) = n + 1 for every n > 1 . This generalizes famous conjectures of Ryser, Brualdi and Stein. Recently, Aharoni et al. proved that g ( n ) ≤ ⌊ 7 4 n ⌋ . We prove that g ( n ) ≤ ⌊ 5 3 n ⌋ .