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Matroids provide an abstract notion of independence and have been extensively
studied in the finite.
In this joint work with Nathan Bowler and Johannes Carmesin
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 size.
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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