## Abstract

We say a matrix is *simple* if it is a (0,1)-matrix with no repeated columns. Given *m* and a *k*×*l* (0,1)-matrix *F* we define forb(*m,F*) as the maximum number of columns in a simple *m*-rowed matrix *A* for which no *k*×*l* submatrix of *A* is a row and column permutation of *F*. In set theory notation, *F* is a forbidden *trace*. For all *k*-rowed *F* (simple or non-simple) Füredi has shown that forb(*m,F*) is *O*(*m*
^{k}). We are able to determine for which *k*-rowed *F* we have that forb(*m,F*) is *O*(*m*
^{k−1}) and for which *k*-rowed *F* we have that forb(*m,F*) is *Θ*(*m*
^{k}).

We need a bound for a particular choice of *F*. Define *D*
_{12} to be the *k*×(2^{k}−2^{k−2}−1) (0,1)-matrix consisting of all nonzero columns on *k* rows that do not have [
^{1}_{1}
] in rows 1 and 2. Let **0** denote the column of *k* 0’s. Define *F*
_{
k
}(*t*) to be the concatenation of **0** with *t*+1 copies of *D*
_{12}. We are able to show that forb(*m,F*
_{
k
}(*t*)) is *Θ*(*m*
^{k−1}). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.

The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed *F* of forb(*m,F*).

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Research supported in part by NSERC.

Research partially supported by NSERC of first author.

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Anstee, R.P., Fleming, B. Linear algebra methods for forbidden configurations.
*Combinatorica* **31, **1 (2011). https://doi.org/10.1007/s00493-011-2595-6

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### Mathematics Subject Classification (2000)

- 05D05
- 05C65
- 05C35