Self-dual uniform matroids on infinite sets

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.