On Kahn's basis conjecture

TL;DR

This paper extends previous results on Rota's basis conjecture to Kahn's basis conjecture, demonstrating that suitable representatives can be chosen in a large portion of the array, with implications for matroids.

Contribution

The authors generalize their earlier findings to Kahn's basis conjecture and matroids, showing that a significant fraction of representatives can be selected to satisfy basis conditions.

Findings

Suitable representatives can be chosen in approximately half of the rows.

Results extend to the setting of Kahn's conjecture and matroids.

The approach builds on and modifies previous methods for Rota's basis conjecture.

Abstract

In 1991, Kahn made the following conjecture. For any $n$-dimensional vector space $V$ and any $n\times n$ array of $n^2$ bases of $V$, it is possible to choose a representative vector from each of these bases in such a way that the representatives from each row form a basis and the representatives from each column also form a basis. Rota's basis conjecture can be viewed as a special case of Kahn's conjecture, where for each column, all the bases in that column are the same. Recently the authors showed that in the setting of Rota's basis conjecture it is possible to find suitable representatives in $\left(1/2-o\left(1\right)\right)n$ of the rows. In this companion note we give a slight modification of our arguments which generalises this result to the setting of Kahn's conjecture. Our results also apply to the more general setting of matroids.