On generalisations of the Aharoni–Pouzet base exchange theorem

Abstract

AbstractThe Greene–Magnanti theorem states that if is a finite matroid, and are bases and is a partition, then there is a partition such that is a base for every . The special case where each 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 and are bases, then there is a bijection between their finite subsets such that is a base for every . In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.