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An excluded minors method for infinite matroids
, 2014
"... The notion of thin sums matroids was invented to extend the notion of representability to nonfinitary matroids. A matroid is tame if every circuitcocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are rep ..."
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The notion of thin sums matroids was invented to extend the notion of representability to nonfinitary matroids. A matroid is tame if every circuitcocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about finite matroids representable over a finite field to theorems about tame thin sums matroids over these fields. We give three examples of this: various characterisations of binary tame matroids and of regular tame matroids, and unique representability of ternary tame matroids.
The ubiquity of Psimatroids
, 2014
"... Solving (for tame matroids) a problem of AignerHorev, Diestel and Postle, we prove that every tame matroid M can be reconstructed from its canonical tree decomposition into 3connected pieces, circuits and cocircuits together with information about which ends of the decomposition tree are used by ..."
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Solving (for tame matroids) a problem of AignerHorev, Diestel and Postle, we prove that every tame matroid M can be reconstructed from its canonical tree decomposition into 3connected pieces, circuits and cocircuits together with information about which ends of the decomposition tree are used by M. For every locally finite graph G, we show that every tame matroid whose circuits are topological circles of G and whose cocircuits are bonds of G is determined by the set Ψ of ends it uses, that is, it is a Ψmatroid. 1
Infinite Gammoids: Minors and Duality
, 2015
"... This sequel to Afzali Borujeni et. al. (2015) considers minors and duals of infinite gammoids. We prove that the class of gammoids defined by digraphs not containing a certain type of substructure, called an outgoing comb, is minorclosed. Also, we prove that finiterank minors of gammoids are gammo ..."
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This sequel to Afzali Borujeni et. al. (2015) considers minors and duals of infinite gammoids. We prove that the class of gammoids defined by digraphs not containing a certain type of substructure, called an outgoing comb, is minorclosed. Also, we prove that finiterank minors of gammoids are gammoids. Furthermore, the topological gammoids of Carmesin (2014) are proved to coincide, as matroids, with the finitary gammoids. A corollary is that topological gammoids are minorclosed. It is a wellknown fact that the dual of any finite strict gammoid is a transversal matroid. The class of strict gammoids defined by digraphs not containing alternating combs, introduced in Afzali Borujeni et. al. (2015), contains examples which are not dual to any transversal matroid. However, we describe the duals of matroids in this class as a natural extension of transversal matroids. While finite gammoids are closed under duality, we construct a strict gammoid that is not dual to any gammoid.
Infinite graphic matroids  Part I
, 2014
"... An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graphlike topological space: that is, a graphlike space in the sense of Thomassen and ..."
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An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graphlike topological space: that is, a graphlike space in the sense of Thomassen and Vella. This extends Tutte’s characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3connected graphic matroid is countable.
On the intersection conjecture for infinite trees of matroids
, 2014
"... Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2separations into finite parts. 1 ..."
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Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2separations into finite parts. 1