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

## 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

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## 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