On a conjecture of Stein

TL;DR

This paper improves the lower bound for the size of partial rainbow matchings in edge-partitioned complete bipartite graphs, advancing towards Stein's conjecture using topological methods.

Contribution

It introduces a topological approach that increases the guaranteed size of partial rainbow matchings from about (1 - 1/e)n to 2/3 n, strengthening previous bounds.

Findings

Established a new lower bound of 2/3 n for partial rainbow matchings.

Used topological methods to improve combinatorial bounds.

Progressed towards proving Stein's conjecture.

Abstract

Stein proposed the following conjecture: if the edge set of $K_{n,n}$ is partitioned into $n$ sets, each of size $n$, then there is a partial rainbow matching of size $n-1$. He proved that there is a partial rainbow matching of size $n(1-\frac{D_n}{n!})$, where $D_n$ is the number of derangements of $[n]$. This means that there is a partial rainbow matching of size about $(1- \frac{1}{e})n$. Using a topological version of Hall's theorem we improve this bound to $\frac{2}{3}n$.