On generalisations of the Aharoni–Pouzet base exchange theorem

Abstract

The Greene–Magnanti theorem states that if (Formula presented.) is a finite matroid, (Formula presented.) and (Formula presented.) are bases and (Formula presented.) is a partition, then there is a partition (Formula presented.) such that (Formula presented.) is a base for every (Formula presented.). The special case where each (Formula presented.) is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite-dimensional vector spaces. Later, he and Aharoni answered this conjecture affirmatively not just for vector spaces but also for infinite matroids. We prove two generalisations of their result. On the one hand, we show that ‘being a singleton’ can be relaxed to ‘being finite’ and this is sharp in the sense that the exclusion of infinite sets is really necessary. In addition, we prove that if (Formula presented.) and (Formula presented.) are bases, then there is a bijection (Formula presented.) between their finite subsets such that (Formula presented.) is a base for every (Formula presented.). In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory. © 2023 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.