Matroid intersection and packing/covering conjectures are true in the class of finitary matroids

Abstract

Given two finite matroids on the same ground set, a celebrated result of Edmonds says that the ground set can be partitioned into two disjoint subsets in a manner that there is a common independent set in both matroids whose intersection with the first subset spans that subset in the first matroid, and whose intersection with the second subset spans that subset in the second matroid. There is a longstanding conjecture regarding the situation of two matroids defined on the same infinite ground set. Infinite matroids were only recently axiomatized in the early 2010s in the work of Bruhn et al., while the conjecture had been proposed during the 1990s for the class of structures that are now called finitary matroids, which are matroids all of whose circuits are finite sets. The packing/covering conjecture, due to Bowler and Carmesin, is a related conjecture in the sense that it is true in the class of all matroids if and only if the matroid intersection conjecture is true in the class of all matroids. Given any family of matroids on the same ground set, the conjecture asks if it is possible to partition the ground set into two disjoint subsets in a way such that the corresponding family of matroids restricted to the first subset admits a packing while the family of matroids contracted to the second subset admits a covering. We prove both of these conjectures in the class of finitary matroids. Our main tool is nonstandard analysis, specifically the technique of iterated nonstandard extensions. Roughly, we first embed any infinite matroid inside a hyperfinite matroid defined on a subset of the nonstandard extension of the original ground set, and we iteratively nonstandardly extend the hyperfinite structure again in order to prove results in the internal universe that can be directly transferred to obtain results about the matroid(s) we started with.