Graphs and Combinatorics

, Volume 30, Issue 6, pp 1325–1349

Forbidden Configurations and Product Constructions

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

DOI: 10.1007/s00373-013-1365-1

Cite this article as:
Anstee, R.P., Koch, C., Raggi, M. et al. Graphs and Combinatorics (2014) 30: 1325. doi:10.1007/s00373-013-1365-1
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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 

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

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