The Extremal Functions of Classes of Matroids of Bounded Branch-Width

Abstract

For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field \(\mathbb{F}\), if M is a minor-closed class of \(\mathbb{F}\)-representable matroids of bounded branch-width, then limn→ ∞ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) - ∆n is periodic when n is sufficiently large and that exM is achieved by a subclass of M of bounded path-width.

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References

  1. [1]

    D. Eppstein: Densities of minor-closed graph families, Electr. Journ. of Comb. 17(1) (2010), Paper R136.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    J. F. Geelen, A. M. H. Gerards and G. Whittle: Branch-width and well-quasiordering in matroids and graphs, J. Combin. Theory Ser. B 84 (2002), 270–290.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. Geelen, B. Gerards and G. Whittle: Excluding a planar graph from GF(q)representable matroids, J. Combin. Theory Ser. B 97 (2007), 971–998.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    J. Geelen, B. Gerards and G. Whittle: On Rota’s conjecture and excluded minors containing large projective geometries, J. Combin. Theory Ser. B 96 (2006), 405–425.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    J. Geelen, B. Gerards and G. Whittle: The Highly Connected Matroids in MinorClosed Classes, Ann. Comb. 19 (2015), 107–123.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    J. Geelen and G. Whittle: Cliques in dense GF(q)-representable matroids, J. Combin. Theory Ser. B 87 (2003), 264–269.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    R. Kapadia and S. Norin: in preparation.

  8. [8]

    W. Mader: Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Ann. 174 (1967), 265–268.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    Z.-X. Song and R. Thomas: The extremal function for K9 minors, J. Combin. Theory Ser. B 96 (2006), 240–252.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Rohan Kapadia.

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Kapadia, R. The Extremal Functions of Classes of Matroids of Bounded Branch-Width. Combinatorica 38, 193–218 (2018). https://doi.org/10.1007/s00493-016-3425-7

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Mathematics Subject Classification (2000)

  • 05B35