Rainbow matchings in bipartite graphs

Given k disjoint matchings in a bipartite graph, a rainbow matching is a matching that contains one edge from each of them. Is it true that any family of k disjoint matchings of size k+1 has a rainbow matching?

Remarks

The conjecture was suggested by Aharoni and Berger as a generalization of the conjecture of Brualdi and Stein on Latin squares, see the survey . Kotlar and Ziv proved that any family of k disjoint matchings of size $$\lfloor \frac{5}{3} k\rfloor$$ has a rainbow matching. Clemens and Ehrenmüller improved the bound to $$\frac{3}{2}k+o(k)$$ , and recently Pokrovskiy improved it to $$k+o(k)$$ , which can be regarded as an asymptotic version of the Aharoni-Berger conjecture. Additional results can be found in . The following matroidal generalization is also open:

Conjecture . Let $$M_1=(S,r_1)$$ and $$M_2=(S,r_2)$$ be two matroids of rank k+1 on S, such that S can be partitioned into k common bases $$B_1, \dots, B_k$$ . Then the family $$B_1, \dots, B_k$$ has a transversal that is independent in both matroids.

There are two related results; the first one is a weaker version of the conjecture, while the second one is a complementary result (its special case involving bipartite graphs is Drisko's Theorem ).

Theorem . Let $$M_1=(S,r_1)$$ and $$M_2=(S,r_2)$$ be two matroids of rank k on S, such that S can be partitioned into k common bases $$B_1, \dots, B_k$$ . Then the family $$B_1, \dots, B_k$$ has a partial transversal of size $$k-\sqrt{k}$$ that is independent in both matroids.

Theorem . Let $$M_1=(S,r_1)$$ and $$M_2=(S,r_2)$$ be two matroids of rank k on S, such that S can be partitioned into 2k-1 common bases $$B_1, \dots, B_{2k-1}$$ . Then the family $$B_1, \dots, B_{2k-1}$$ has a partial transversal of size $$k$$ that is a common base.