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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
R. Aharoni and E. Berger: Menger’s theorem for infinite graphs, Invent. math. 176 (2009), 1–62.
R. Aharoni and C. Thomassen: Infinite, highly connected digraphs with no two arc-disjoint spanning trees, J. Graph Theory 13 (1989), 71–74.
R. Aharoni and R. Ziv: The intersection of two infinite matroids, J. London Math. Soc. 58 (1998), 513–525.
E. Aigner-Horev, J. Carmesin and J. Fröhlich: Infinite matroid union, preprint 2012, available at http://arxiv.org/pdf/1111.0602v2.
E. Aigner-Horev, J. Carmesin and J. Fröhlich: On the intersection of infinite matroids, preprint 2012, available at http://arxiv.org/pdf/1111.0606v2.
M. A. Armstrong: Basic Topology, Springer-Verlag, 1983.
H. Bruhn and R. Diestel: Infinite matroids in graphs, Infinite Graph Theory special volume of Discrete Math 311 (2011), 1461–1471.
H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh and P. Wollan: Axioms for infinite matroids, Adv. Math 239 (2013), 18–46
H. Bruhn and P. Wollan: Finite connectivity in infinite matroids, European J. of Combin. 33 (2012), 1900–1912.
R. Christian: Infinite graphs, graph-like spaces, B-matroids, PhD thesis, University of Waterloo, 2010.
R. Diestel: Graph Theory (4th edition). Springer-Verlag, 2010. Electronic edition available at: http://diestel-graph-theory.com/index.html.
C. St. J. A. Nash-Williams: Infinite graphs—a survey, J. Combin. Theory 3 (1967), 286–301.
J. Oxley: Matroid Theory, Oxford University Press, 1992.
About this article
Cite this article
Bowler, N., Carmesin, J. Matroid intersection, base packing and base covering for infinite matroids. Combinatorica 35, 153–180 (2015). https://doi.org/10.1007/s00493-014-2953-2
Mathematics Subject Classification (2010)