Abstract
We consider the problem of determining m n , the number of matroids on n elements. The best known lower bound on m n is due to Knuth (1974) who showed that loglogm n is at least n − 3/2logn − O(1). On the other hand, Piff (1973) showed that loglogm n ≤ n − logn + loglogn + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth’s bound.
We show that this is indeed the case, and prove an upper bound on loglogm n that is within an additive 1+o(1) term of Knuth’s lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of stable sets in the Johnson graph to give a compressed representation of matroids.
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References
N. Alon and F. R. K. Chung: Explicit construction of linear sized tolerant networks, Discrete Math. 72 (1988), 15–19.
N. Alon, J. Balogh, R. Morris and W. Samotij: Counting sum-free sets in Abelian groups, Israel Journal of Mathematics 199 (2014), 309–344.
N. Alon and J. H. Spencer: The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., Hoboken, NJ, third edition, 2008, with an appendix on the life and work of Paul Erdős.
J. E. Blackburn, H. H. Crapo and D. A. Higgs: A catalogue of combinatorial geometries, Math. Comp., 27:155–166; addendum, ibid. 27 (1973), no. 121, loose microfiche suppl. A12–G12, 1973.
J. E. Bonin: Sparse paving matroids, basis-exchange properties, and cyclic flats, arXiv:1011.1010v1, 2011.
A. E. Brouwer and T. Etzion: Some new constant weight codes, Advances in Mathematics of Communications 5 (2011), 417–424.
A. E. Brouwer, A. M. Cohen and A. Neumaier: Distance-regular graphs, Springer, 1989.
A. E. Brouwer and W. H. Haemers: Spectra of graphs, Universitext. Springer, New York, 2012.
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.
J. Geelen and P. J. Humphries: Rota’s basis conjecture for paving matroids, SIAM J. Discrete Math., 20(4):1042–1045 (electronic), 2006.
R. L. Graham and N. J. A. Sloane: Lower bounds for constant weight codes, IEEE Trans. Inform. Theory 26 (1980), 37–43.
W. H. Haemers: Eigenvalue techniques in design and graph theory, volume 121 of Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam, 1980. Dissertation, Technische Hogeschool Eindhoven, Eindhoven, 1979.
M. Jerrum: Two remarks concerning balanced matroids, Combinatorica 26 (2006), 733–742.
D. J. Kleitman and K. J. Winston: On the number of graphs without 4-cycles, Discrete Math. 41 (1982), 167–172.
D. E. Knuth: The asymptotic number of geometries, J. Combinatorial Theory Ser. A 16 (1974), 398–400.
J. P. S. Kung: Matroids, in: Handbook of algebra, Vol. 1, 157–184. North-Holland, Amsterdam, 1996.
D. Mayhew, M. Newman, D. Welsh and G. Whittle: On the asymptotic proportion of connected matroids, European J. Combin. 32 (2011), 882–890.
D. Mayhew, M. Newman and G. Whittle: Is the missing axiom of matroid theory lost forever?, arXiv:1204.3365, 2012.
D. Mayhew and G. F. Royle: Matroids with nine elements, J. Combin. Theory Ser. B 98 (2008), 415–431.
D. Mayhew and D. Welsh: On the number of sparse paving matroids, http://homepages.ecs.vuw.ac.nz/~mayhew/Publications/MW.pdf, 2010.
C. Merino, S. D. Noble, M. Ramírez-Ibáñez and R. V.-Flores: On the structure of the h-vector of a paving matroid, Eur. J. Comb. 33 (2012), 1787–1799.
J. Oxley: Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, second edition, 2011.
R. A. Pendavingh and J. G. van der Pol: Counting matroids in minor-closed classes, arXiv:1302.1315v3, 2013.
M. J. Piff: An upper bound for the number of matroids, J. Combinatorial Theory Ser. B 14 (1973), 241–245.
M. J. Piff and D. J. A. Welsh: The number of combinatorial geometries, Bull. London Math. Soc. 3 (1971), 55–56.
A. Schrijver: Combinatorial optimization. Polyhedra and efficiency. Vol. B, volume 24 of Algorithms and Combinatorics, Springer-Verlag, Berlin, 2003. Matroids, trees, stable sets, Chapters 39–69.
D. J. A. Welsh: Matroid theory, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976. L. M. S. Monographs, No. 8.
H. Whitney: On the Abstract Properties of Linear Dependence, Amer. J. Math. 57 (1935), 509–533.
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This research has been supported by the Netherlands Organisation for Scientific Research (NWO) grant 639.022.211 and 613.001.211.
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Bansal, N., Pendavingh, R.A. & van der Pol, J.G. On the number of matroids. Combinatorica 35, 253–277 (2015). https://doi.org/10.1007/s00493-014-3029-z
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DOI: https://doi.org/10.1007/s00493-014-3029-z