Graphs and Combinatorics

, Volume 30, Issue 6, pp 1325–1349 | Cite as

Forbidden Configurations and Product Constructions

  • Richard P. Anstee
  • Christina Koch
  • Miguel Raggi
  • Attila Sali
Original Paper
  • 131 Downloads

Abstract

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let \({\|A\|}\) denote the number of columns of A. We define \({{\rm forb}(m, F) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration F. We extend this to a family \({\mathcal{F} = \{F_1, F_2, \ldots , F_t\}}\) and define \({{\rm forb}(m, \mathcal{F}) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration \({F \in \mathcal{F}\}}\) . We consider products of matrices. Given an m1 × n1 matrix A and an m2 × n2 matrix B, we define the productA × B as the (m1m2) × n1n2 matrix whose columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let Ik denote the k × k identity matrix, let \({I_k^{c}}\) denote the (0,1)-complement of Ik and let Tk denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I2 × I2, T2 × T2}) is \({\Theta(m^{3/2})}\) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)-simple matrices. For two matrices F, P, where P is m-rowed, let \({f(F, P) = {\rm max}_{A} \{\|A\| \,:\,A}\) is m-rowed submatrix of P with no configuration F}. We establish f(I2 × I2, Im/2 × Im/2) is \({\Theta(m^{3/2})}\) whereas f(I2 × T2, Im/2 × Tm/2) and f(T2 × T2, Tm/2 × Tm/2) are both \({\Theta(m)}\). Additional results are obtained. One of the results requires extensive use of a computer program. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.

Keywords

VC-dimension Forbidden configurations Trace Patterns Products 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anstee, R.P.: A survey of forbidden configurations results. Elec. J. Comb. 20, DS20 (2013)Google Scholar
  2. 2.
    Anstee R.P., Fleming B.: Two refinements of the bound of Sauer, Perles and Shelah and Vapnik and Chervonenkis. Discrete Math. 310, 3318–3323 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anstee, R.P., Griggs, J.R., Sali, A. Small forbidden configurations. Graphs Comb. 13, 97–118 (1997)Google Scholar
  4. 4.
    Anstee R.P., Sali A.: Small forbidden configurations IV. Combinatorica 25, 503–518 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anstee, R.P., Fleming, B., Füredi, Z., Sali, A.: Color critical hypergraphs and forbidden configurations, proceedings of EuroComb 2005, Berlin, Germany. Discrete Math. Theor. Comput. Sci. AE, 117–122 (2005)Google Scholar
  6. 6.
    Balogh J., Bollobás B.: Unavoidable traces of set systems. Combinatorica 25, 633–643 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Balogh J., Bollobás B., Morris R.: Hereditary properties of partitions, ordered graphs and ordered hypergraphs. Eur. J. Comb. 8, 1263–1281 (2006)CrossRefGoogle Scholar
  8. 8.
    Erdős P.: On extremal problems of graphs and generalized graphs. Israel J. Math. 2, 183–190 (1964)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Erdős P., Simonovits M.: A limit theorem in graph theory. Studia Sci. Math. Hung. 1, 51–57 (1966)Google Scholar
  10. 10.
    Erdős P., Stone A.H.: On the structure of linear graphs. Bull. AMS 52, 1089–1091 (1946)Google Scholar
  11. 11.
    Füredi Z.: An upper bound on Zarankiewicz problem. Comb. Probab. Comput. 5, 29–33 (1996)CrossRefMATHGoogle Scholar
  12. 12.
    Füredi Z., Hajnal P.: Davenport–Schinzel theory of matrices. Discrete Math. 103, 233–251 (1992)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klazar M., Marcus A.: Extensions of the linear bound in the Füredi–Hajnal conjecture. Adv. Appl. Math. 38, 258–266 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kővari, T., Sós, V., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math 3(1954), 50–57.Google Scholar
  15. 15.
    Marcus A., Tardos G.: Excluded permutation matrices and the Stanley Wilf conjecture. J. Combin. Theory Ser. A 107, 153–160 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
  17. 17.
    Tardos G.: On 0–1 matrices and small excludedsubmatrices. J. Combin. Theory Ser. A 111, 266–288MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  • Richard P. Anstee
    • 1
  • Christina Koch
    • 1
  • Miguel Raggi
    • 1
  • Attila Sali
    • 2
  1. 1.Mathematics DepartmentThe University of British ColumbiaVancouverCanada
  2. 2.Alfréd Rényi Institute, Hungarian Academy of SciencesBudapestHungary

Personalised recommendations