Linear algebra methods for forbidden configurations

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×(2k−2k−2−1) (0,1)-matrix consisting of all nonzero columns on k rows that do not have [ 11 ] 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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Correspondence to Richard P. Anstee.

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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