Representation of large matchings in bipartite graphs

Abstract

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenm{\"u}ller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \le \lceil \frac{3}{2}n \rceil+1$.