On the intersection of infinite matroids

TL;DR

This paper connects the infinite matroid intersection conjecture to the infinite Menger theorem, proving the conjecture for certain classes of nearly finitary matroids and applying it to specific graph structures.

Contribution

It demonstrates that the infinite matroid intersection conjecture implies the infinite Menger theorem and proves the conjecture for nearly finitary and dual nearly finitary matroids.

Findings

The infinite matroid intersection conjecture implies the infinite Menger theorem.

The conjecture holds for nearly finitary and dual nearly finitary matroids.

It applies to finite-cycle matroids of certain 2-connected, locally finite graphs.

Abstract

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger. We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids. In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.