## Self-dual uniform matroids on infinite sets

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- by Nathan Bowler and Stefan Geschke PDF
- Proc. Amer. Math. Soc.
**144**(2016), 459-471 Request permission

## 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
- MR Author ID: 681801
- Email: stefan.geschke@uni-hamburg.de
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 459-471 - MSC (2010): Primary 05B35; Secondary 03E35, 03E25, 03E50
- DOI: https://doi.org/10.1090/proc/12667
- MathSciNet review: 3430826