Odd circuits in dense binary matroids

Abstract

We show that, for each real number α>0 and odd integer k≥5, there is an integer c such that, if M is a simple binary matroid with |M|≥α2r(M) and with no k-element circuit, then M has critical number at most c. The result is an easy application of a regularity lemma for finite abelian groups due to Green.

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References

  1. [1]

    H. H. Crapo and G.-C. Rota: On the Foundations of Combinatorial Theory: Combinatorial Geometries, Preliminary edition, MIT Press, Cambridge, 1970.

    Google Scholar 

  2. [2]

    P. Erdős and M. Simonovits: On a valence problem in extremal graph theory, Discrete Math. 5 (1973), 323–334.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    H. Furstenberg and Y. Katznelson: IP-sets, Szemerćdi’s Theorem and Ramsey Theory, Bull. Amer. Math. Soc. (N.S.) 14 (1986), 275–278.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    B. Green: A Szemeredi-type regularity lemma in abelian groups, with applications, Geometric & Functional Analysis GAFA 15 (2005), 340–376.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    J. G. Oxley: Matroid Theory, Oxford University Press, New York (2011).

    Google Scholar 

  6. [6]

    J. G. Oxley: The contributions of Dominic Welsh to matroid theory, in: Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh, Oxford University Press, 2007.

    Google Scholar 

  7. [7]

    T. C. Tao and V. H. Vu: Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge (2006).

    Google Scholar 

  8. [8]

    C. Thomassen: On the chromatic number of pentagon-free graphs of large minimum degree, Combinatorica 27 (2007), 241–243.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Peter Nelson.

Additional information

This research was partially supported by a grant from the Office of Naval Research [N00014-10-1-0851].

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Geelen, J., Nelson, P. Odd circuits in dense binary matroids. Combinatorica 37, 41–47 (2017). https://doi.org/10.1007/s00493-015-3237-1

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Mathematics Subject Classification (2000)

  • 05B35