Matroid intersection, base packing and base covering for infinite matroids

TL;DR

This paper explores how core theorems of finite matroid theory, such as base packing, covering, and intersection, can be extended to infinite matroids, revealing their equivalences and simplifying proofs.

Contribution

It establishes the equivalence of several conjectures in infinite matroid theory and provides new, simplified proofs for finite cases and known infinite cases.

Findings

Several conjectures are equivalent in infinite matroid theory.

New proofs for finite matroid theorems are derived from these equivalences.

Simplified proofs extend to some known infinite matroid cases.

Abstract

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and simplify the proofs of, some cases where these conjectures were known to be true.