Base Partition for Mixed Families of Finitary and Cofinitary Matroids


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.


  1. [1]

    E. AIGNER-HOREV, J. CARMESIN and J.-O. FRÖHLICH: On the intersection of infinite matroids, Discrete Math. 1 (2018), 1582–1596.

    MathSciNet  Article  Google Scholar 

  2. [2]

    S. H. A. BORUJENI and N. BOWLER: Thin sums matroids and duality, Advances in Mathematics 1 (2015), 1–29.

    MathSciNet  Article  Google Scholar 

  3. [3]

    N. BOWLER: Infinite matroids, Habilitation thesis, University Hamburg, University of Hamburg, 2014.

    Google Scholar 

  4. [4]

    N. BOWLER and J. CARMESIN: The ubiquity of psi-matroids, arXiv preprint arXiv: 1304.6973, 2013.

    Google Scholar 

  5. [5]

    N. BOWLER and J. CARMESIN: Matroid intersection, base packing and base covering for infinite matroids, Combinatorial 1 (2015), 153–180.

    MathSciNet  Article  Google Scholar 

  6. [6]

    N. BOWLER, J. CARMESIN and R. CHRISTIAN: Infinite graphic matroids, Combinatorial 1 (2018), 305–339.

    MathSciNet  Article  Google Scholar 

  7. [7]

    N. BOWLER and STEFAN GESCHKE: Self-dual uniform matroids on infinite sets, Proceedings of the American Mathematical Society 1 (2016), 459–471.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    H. BRUHN, R. DIESTEL, M. KRIESELL, R. PENDAVINGH and P. WOLLAN: Axioms for infinite matroids, Advances in Mathematics 1 (2013), 18–46.

    MathSciNet  Article  Google Scholar 

  9. [9]

    E. K. VAN DOUWEN: The integers and topology, in: Handbook of set-theoretic topology, 111–167. Elsevier, 1984.

    Google Scholar 

  10. [10]

    J. EDMONDS and D. R. FULKERSON: Transversals and matroid partition, J. Res. Nat. Bur. Standards Sect. B 1 (1965), 147–153.

    MathSciNet  Article  Google Scholar 

  11. [11]

    J. ERDE, P. GOLLIN, A. JOÓ, P. KNAPPE and M. PITZ: A Cantor-Bernstein-type theorem for spanning trees in infinite graphs, arXiv:1907.09338, 2019.

    Google Scholar 

  12. [12]

    D. A. HIGGS: Matroids and duality, Colloq. Math. 1 (1969), 215–220.

    MathSciNet  Article  Google Scholar 

  13. [13]

    J. OXLEY: Infinite matroids, in: Matroid applications, volume 40 of Encyclopedia Math. Appl., 73–90. Cambridge Univ. Press, Cambridge, 1992.

    Google Scholar 

  14. [14]

    J. E. VAUGHAN: Small uncountable cardinals and topology, in: Open problems in topology, 195–218. North-Holland, Amsterdam, 1990.

    Google Scholar 

Download references

Author information



Corresponding authors

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Erde, J., Gollin, J.P., Joó, A. et al. Base Partition for Mixed Families of Finitary and Cofinitary Matroids. Combinatorica 41, 31–52 (2021).

Download citation

Mathematics Subject Classification (2010)

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