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Axioms for infinite matroids
"... We give axiomatic foundations for nonfinitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966. ..."
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We give axiomatic foundations for nonfinitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.
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 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