On generalisations of the Aharoni-Pouzet base exchange theorem

TL;DR

This paper extends the Greene-Magnanti base exchange theorem to infinite matroids, proving new generalizations that relax the singleton condition to finite sets and establish a bijection between finite subsets of bases, all with elementary proofs.

Contribution

It introduces two new generalizations of the base exchange theorem for infinite matroids, including a finite set relaxation and a finite subset bijection, with elementary proof methods.

Findings

Relaxation from singleton to finite sets is sharp.

Existence of a bijection between finite subsets of bases ensuring base exchange.

Elementary proofs avoid infinite matching theory.

Abstract

The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $ 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 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 the exclusion of infinite sets is really necessary. On the other hand, we prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $ between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.