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Self-dual uniform matroids on infinite sets


Authors: Nathan Bowler and Stefan Geschke
Journal: Proc. Amer. Math. Soc. 144 (2016), 459-471
MSC (2010): Primary 05B35; Secondary 03E35, 03E25, 03E50
Published electronically: October 7, 2015
MathSciNet review: 3430826
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Abstract: We extend the notion of a uniform matroid to the infinitary case and construct, using weak fragments of Martin's Axiom, self-dual uniform matroids on infinite sets. In 1969, Higgs showed that, assuming the Generalised Continuum Hypothesis (GCH), any two bases of a fixed matroid have the same size. We show that this cannot be proved from the usual axioms of set theory, ZFC, alone: in fact, we show that it is consistent with ZFC that there is a uniform self-dual matroid with two bases of different sizes.

Self-dual uniform matroids on infinite sets also provide examples of infinitely connected matroids, answering a question of Bruhn and Wollan under additional set-theoretic assumptions. While we do not know whether the existence of a self-dual uniform matroid on an infinite set can be proved in ZFC alone, we show that ZF, Zermelo-Fraenkel Set Theory without the Axiom of Choice, is not enough. Finally, we observe that there is a model of set theory in which GCH fails while any two bases of a matroid have the same size. This answers a question of Higgs.


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Additional Information

Nathan Bowler
Affiliation: Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
Email: N.Bowler1729@gmail.com

Stefan Geschke
Affiliation: Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
Email: stefan.geschke@uni-hamburg.de

DOI: https://doi.org/10.1090/proc/12667
Received by editor(s): February 28, 2014
Received by editor(s) in revised form: October 6, 2014
Published electronically: October 7, 2015
Additional Notes: We thank Johannes Carmesin for many fruitful discussions on the subject of this article.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society