Monday, November 2, 3pm ET (8pm GMT, 9am Tue NZDT)
Attila Joó, University of Hamburg
The Matroid Intersection Conjecture of Nash-Williams
Rado initiated a program in the 1960s to find the “right” infinite generalization of the matroid concept. The results of Higgs and Oxley and more recently Bruhn et al. led eventually to a positive answer for Rado’s question. One of the most important open problems in the theory of infinite matroids is the Matroid Intersection Conjecture of Nash-Williams which is a structural infinite generalization of the well-known Intersection Theorem of Edmonds. It says that if $M$ and $N$ are (finitary) matroids on the common edge set $E$, then they admit a common independent set $I$ that has a bipartition $I=I_M \cup I_N$ with $cl_M(I_M) \cup cl_N(I_N)=E$. The restriction of the conjecture to partition matroids (known as ‘König’s Theorem for infinite bipartite graphs’) was proven by Aharoni, Nash-Williams and Shelah and is a deep result in infinite matching theory. In the main part of the talk we give a proof overview of our partial result which decides affirmatively the conjecture whenever $E$ is countable. Finally we reveal an unpublished conjecture of Aharoni about the intersection of more than two matroids which is wide open even for three finite matroids.