On the Number of Bases of Almost All Matroids



For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that
$$\Omega (1/n) \leqslant 1 - b(M) \leqslant O(log{(n)^3})asn \to \infty $$
for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a Uk, 2k-minor, whenever kO(log(n)), (2) have girth ≥ Ω(log(n)), (3) have Tutte connectivity \( \geqslant \Omega (\sqrt {\log (n)} )\), and (4) do not arise as the truncation of another matroid.

Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.

Mathematics Subject Classification (2000)

05B35 05A16 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. E. Brouwer, A.M. Cohen and A. Neumaier: Distance-Regular Graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Berlin Heidelberg, 1989.Google Scholar
  2. [2]
    N. Bansal, R. A. Pendavingh and J. G. van der Pol: On the number of matroids, Combinatorica 35 (2015), 253–277.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    F. R. K. Chung, R. L. Graham, P. Frankl and J. B. Shearer: Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A 43 (1986), 23–37.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    B. Cloteaux: Approximating the number of bases for almost all matroids, Congres-sus Numerantium 202 (2010), 149–153.MathSciNetMATHGoogle Scholar
  5. [5]
    H. H. Crapo and G.-C. Rota: On the foundations of combinatorial theory: Combinatorial geometries, the M.I.T. Press, Cambridge, Mass.-London, preliminary edition, 1970.MATHGoogle Scholar
  6. [6]
    W. Critchlow: Minors of asymptotically almost all sparse paving matroids, preprint, available on arXiv:1605.02414, 2016.Google Scholar
  7. [7]
    J. Geelen, B. Gerards and G. Whittle: Solving Rota’s conjecture, Notices of the AMS 61 (2014).Google Scholar
  8. [8]
    R. L. Graham and N. J. A. Sloane: Lower bounds for constant weight codes, IEEE Trans. Inform. Theory 26 (1980), 37–43.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    S. Jukna: Extremal combinatorics: with applications in computer science, Springer, 2011.CrossRefMATHGoogle Scholar
  10. [10]
    D. E. Knuth: The asymptotic number of geometries, J. Combin. Theory Ser. A 16 (1974), 398–400.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. J. Kleitman and K. J. Winston: On the number of graphs without 4-cycles, Discrete Math. 41 (1982), 167–172.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    D. Mayhew, M. Newman and G. Whittle: On excluded minors for real-representability, J. Combin. Theory Ser. B 99 (2009), 685–689.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. Mayhew, M. Newman, D. Welsh and G. Whittle: On the asymptotic pro-portion of connected matroids, European J. Combin. 32 (2011), 882–890.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    P. Nelson: Almost all matroids are non-representable, Preprint, avaible on arXiv:1605.04288v2, 2016.Google Scholar
  15. [15]
    J. Oxley, C. Semple, L. Warshauer and D. Welsh: On properties of almost all matroids, Adv. Appl. Math. 50 (2013), 115–124.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Oxley: Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, second edition, 2011.CrossRefMATHGoogle Scholar
  17. [17]
    R. Pendavingh and J. van der Pol: Counting matroids in minor-closed classes, J. of Combin. Theory Ser. B 111 (2015), 126–147.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    R. Pendavingh and J. van der Pol: On the number of matroids compared to the number of sparse paving matroids, Electron. J. Combin. 22 (2015), Paper 2.51.Google Scholar
  19. [19]
    W. Samotij: Counting independent sets in graphs, Eur. J. Combin. 48 (2015), 5–18.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Eindhoven University of TechnologyEindhoventhe Netherlands
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations