# Matroid intersection, base packing and base covering for infinite matroids

Original Paper

First Online:

## 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.

## Mathematics Subject Classification (2010)

05C63 05B35 05B40## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R. Aharoni and E. Berger: Menger’s theorem for infinite graphs,
*Invent. math.***176**(2009), 1–62.CrossRefzbMATHMathSciNetGoogle Scholar - [2]R. Aharoni and C. Thomassen: Infinite, highly connected digraphs with no two arc-disjoint spanning trees,
*J. Graph Theory***13**(1989), 71–74.CrossRefzbMATHMathSciNetGoogle Scholar - [3]R. Aharoni and R. Ziv: The intersection of two infinite matroids,
*J. London Math. Soc.***58**(1998), 513–525.CrossRefMathSciNetGoogle Scholar - [4]E. Aigner-Horev, J. Carmesin and J. Fröhlich: Infinite matroid union, preprint 2012, available at http://arxiv.org/pdf/1111.0602v2.Google Scholar
- [5]E. Aigner-Horev, J. Carmesin and J. Fröhlich: On the intersection of infinite matroids, preprint 2012, available at http://arxiv.org/pdf/1111.0606v2.Google Scholar
- [6]
- [7]H. Bruhn and R. Diestel: Infinite matroids in graphs,
*Infinite Graph Theory special volume of Discrete Math***311**(2011), 1461–1471.zbMATHMathSciNetGoogle Scholar - [8]H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh and P. Wollan: Axioms for infinite matroids,
*Adv. Math***239**(2013), 18–46CrossRefzbMATHMathSciNetGoogle Scholar - [9]H. Bruhn and P. Wollan: Finite connectivity in infinite matroids,
*European J. of Combin.***33**(2012), 1900–1912.CrossRefzbMATHMathSciNetGoogle Scholar - [10]R. Christian:
*Infinite graphs, graph-like spaces, B-matroids*, PhD thesis, University of Waterloo, 2010.Google Scholar - [11]R. Diestel:
*Graph Theory*(4th edition). Springer-Verlag, 2010. Electronic edition available at: http://diestel-graph-theory.com/index.html.CrossRefGoogle Scholar - [12]C. St. J. A. Nash-Williams: Infinite graphs—a survey,
*J. Combin. Theory***3**(1967), 286–301.CrossRefzbMATHMathSciNetGoogle Scholar - [13]

## Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014