Base Partition for Mixed Families of Finitary and Cofinitary Matroids

Abstract

Let \(\mathcal{M} = ({M_i}:i \in K)\) be a finite or infinite family consisting of matroids on a common ground set E each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each Mi, which covers the set E, and also a collection of bases which are pairwise disjoint, then there is a collection of bases which partition E. We also show that the failure of this Cantor-Bernstein-type statement for arbitrary matroid families is consistent relative to the axioms of set theory ZFC.

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Correspondence to Joshua Erde or J. Pascal Gollin or Attila Joó or Paul Knappe or Max Pitz.

Additional information

The second author was supported by the Institute for Basic Science (IBS-R029-C1).

The third author would like to thank the generous support of the Alexander von Humboldt Foundation and NKFIH OTKA-129211.

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Erde, J., Gollin, J.P., Joó, A. et al. Base Partition for Mixed Families of Finitary and Cofinitary Matroids. Combinatorica 41, 31–52 (2021). https://doi.org/10.1007/s00493-020-4422-4

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Mathematics Subject Classification (2010)

  • 05B35
  • 05B40
  • 05C63
  • 03E35