Infinite Matroids and Determinacy of Games

Abstract

Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that the class of planar infinite matroids does not have a universal matroid.The existence of these matroids has a connection to Set Theory in that it corresponds to the Determinacy of certain games. To show that our construction gives matroids, we introduce a new very simple axiomatization of the class of countable tame matroids.

Axioms for infinite matroids

“…The duals of these matroids have as their circuits the bonds of G that have no end of Ψ in their closure. See Bowler and Carmesin [7] for details.…”

“…As before, Corollary 2.3 and Theorem 2.4 are but the extreme cases of a more subtle duality of planar graphs, which is reflected by the Ψ-matroids indicated at the end of Section 2.2. See [7,23] for details. More on infinite graphic matroids can be found in [12].…”

“…Since T ∞ does not contain the Bean graph as a subdivision, they are the circuits of a matroid M T ∞ on the edge set of T ∞ , by Theorem 2.5. 7 To show that every cocircuit is infinite we borrow Lemma 3.11 from Section 3, which says that a circuit and a cocircuit never meet in exactly one element. Since for any finite edge set F in T ∞ it is easy to find a double ray meeting F in exactly one edge, we deduce that F cannot be a cocircuit.…”

“…A matroid, thinly representable or not, is tame if every circuit meets every cocircuit only finitely. Tame matroids are substantially easier to handle than arbitrary matroids, and have more pleasant properties [7]. Forbidden minor characterizations extend readily from finite to tame matroids [6]; for example, a tame matroid is thinly representable over F 2 if and only if it does not have U 2,4 as a minor.…”

“…The class of tame matroids is closed under taking minors (as well as, by definition, under duality), so the tame matroids also solve Rado's original problem. See Bowler and Carmesin [6,7,9,11] for more.…”