Colloquium: Jerzy Wojciechowski, West Virginia University
Title: The Matroid Intersection Conjecture of Nash-Williams
Abstract: The Matroid Intersection Conjecture of Nash-Williams (for finitary matroids on a countable ground-set) has been proved by Attila Joó (Hamburg University, Germany). I plan to present a sketch of the proof.
It has been one of the most important open problem in infinite matroid theory for decades. It contains as a special case the generalization of Menger’s theorem to infinite graphs conjectured by Erdős and proved by Aharoni and Berger. It is a generalization of the Matroid Intersection Theorem of Edmonds.
A typical example of a finitary matroid is a subset of a vector space over some field together with the structure of linear independence. The Conjecture says that given two finitary matroids on the same ground-set there exists a subset that is independent in both matroids and that can be partitioned into two sets such that each element of the ground-set is either spanned by the first set in the first matroid or by the second set in the second matroid.