Combinatorica

, Volume 37, Issue 1, pp 41–47 | Cite as

Odd circuits in dense binary matroids

Original Paper

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.

Mathematics Subject Classification (2000)

05B35 

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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.MATHGoogle Scholar
  2. [2]
    P. Erdős and M. Simonovits: On a valence problem in extremal graph theory, Discrete Math. 5 (1973), 323–334.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    B. Green: A Szemeredi-type regularity lemma in abelian groups, with applications, Geometric & Functional Analysis GAFA 15 (2005), 340–376.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J. G. Oxley: Matroid Theory, Oxford University Press, New York (2011).CrossRefMATHGoogle 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.MATHGoogle Scholar
  7. [7]
    T. C. Tao and V. H. Vu: Additive Combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge (2006).CrossRefMATHGoogle Scholar
  8. [8]
    C. Thomassen: On the chromatic number of pentagon-free graphs of large minimum degree, Combinatorica 27 (2007), 241–243.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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