Aigner-Horev, Elad; Carmesin, Johannes; Fröhlich, Jan-Oliver On the intersection of infinite matroids. (English) Zbl 1384.05061 Discrete Math. 341, No. 6, 1582-1596 (2018). Summary: We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by R. Aharoni and E. Berger [Invent. Math. 176, No. 1, 1–62 (2009; Zbl 1216.05092)]. We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids. In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays. Cited in 8 Documents MSC: 05B35 Combinatorial aspects of matroids and geometric lattices 05C63 Infinite graphs 52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) Keywords:infinite matroids; infinite graphs; matroid intersection Citations:Zbl 1216.05092 PDFBibTeX XMLCite \textit{E. Aigner-Horev} et al., Discrete Math. 341, No. 6, 1582--1596 (2018; Zbl 1384.05061) Full Text: DOI arXiv References: [1] Aharoni, R.; Berger, E., Menger’s theorem for infinite graphs, Invent. Math., 176, 1-62 (2009) · Zbl 1216.05092 [2] Aharoni, R.; Ziv, R., The intersection of two infinite matroids, J. London Math. Soc., 58, 513-525 (1998) · Zbl 0922.05017 [3] E. Aigner-Horev, J. Carmesin, J. Fröhlich, Infinite matroid union, arXiv:1111.0602v2; E. Aigner-Horev, J. Carmesin, J. Fröhlich, Infinite matroid union, arXiv:1111.0602v2 [4] Armstrong, M. A., Basic Topology (1983), Springer-Verlag · Zbl 0514.55001 [5] Bowler, Nathan; Carmesin, Johannes, Matroid intersection, base packing and base covering for infinite matroids, Combinatorica, 35, 2, 153-180 (2015) · Zbl 1349.05051 [6] Bruhn, Henning; Diestel, Reinhard, Infinite matroids in graphs, Discrete Math., 311, 15, 1461-1471 (2011) · Zbl 1231.05055 [7] Bruhn, Henning; Diestel, Reinhard; Kriesell, Matthias; Pendavingh, Rudi; Wollan, Paul, Axioms for infinite matroids, Adv. Math., 239, 18-46 (2013) · Zbl 1279.05013 [8] Christian, R., Infinite Ggraphs, Ggraph-like Spaces and B-matroids (2010), University of Waterloo, (Ph.D. thesis) [9] Diestel, R., Graph Theory (2010), Springer-Verlag, Electronic edition available at: http://diestel-graph-theory.com/index.html · Zbl 1204.05001 [10] Higgs, D. A., Infinite graphs and matroids, (Recent Progress in Combinatorics, Proceedings Third Waterloo Conference on Combinatorics (1969), Academic Press), 245-253 · Zbl 0195.54302 [11] Oxley, J., Matroid Theory (1992), Oxford University Press · Zbl 0784.05002 [12] Schrijver, A., Combinatorial Optimization - Polyhedra and Efficiency - Volume B (2003), Springer-Verlag · Zbl 1041.90001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.