# 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).

This is a preview of subscription content, access via your institution.

## References

1. [1]

R. Ahlswede and L. H. Khatchatrian: Counterexample to the Frankl-Pach conjecture for uniform, dense families; Combinatorica 17(2) (1997), 299–301.

2. [2]

R. P. Anstee: A Survey of Forbidden Configurations results, http://www.math.ubc.ca/~anstee.

3. [3]

R. P. Anstee and B. Fleming: Two refinements of the bound of Sauer, Perles and Shelah, and of Vapnik and Chervonenkis; Discrete Math. 310(23) (2010), 3318–3323.

4. [4]

R. P. Anstee, B. Fleming, Z. Füredi and A. Sali: Color critical hypergraphs and forbidden configurations, in: Proceedings of EuroComb’ 05, (2005), pp. 117–122, Berlin, Germany.

5. [5]

R. P. Anstee, J. R. Griggs and A. Sali: Small forbidden configurations, Graphs and Combinatorics 13 (1997), 97–118.

6. [6]

R. P. Anstee and A. Sali: Small forbidden configurations IV: The 3 rowed case; Combinatorica 25(5) (2005), 503–518.

7. [7]

J. Balogh and B. Bollobás: Unavoidable traces of set systems, Combinatorica 25(6) (2005), 633–643.

8. [8]

J. Balogh, P. Keevash and B. Sudakov: Disjoint representability of sets and their complements, J. Combin. Th. Ser. B 95 (2005), 12–28.

9. [9]

P. Frankl and J. Pach: On disjointly representable sets, Combinatorica 4(1) (1984), 39–45.

10. [10]

Z. Füredi: private communication, 1983.

11. [11]

Z. Füredi and J. Pach: Traces of finite sets: Extremal problems and geometric applications; in: Extremal Problems for Finite Sets (Visegrád, 1991), Bolyai Society Mathematical Studies, v. 3, (1994), pp. 251–282, János Bolyai Math. Soc., Budapest.

12. [12]

A. Marcus and G. Tardos: Excluded permutation matrices and the Stanley Wilf Conjecture, J. Combin. Th. Ser. A 107 (2004), 153–160.

13. [13]

D. Mubayi and J. Zhao: On the VC-dimension of uniform hypergraphs, J. Algebraic Combinatorics 25 (2007), 101–110.

14. [14]

D. Mubayi and J. Zhao: Forbidding complete hypergraphs as traces, Graphs and Combinatorics 23 (2007), 667–679.

15. [15]

B. Patkós: Traces of uniform families of sets, Elec. J. Combinatorics 16 (2009), N8.

16. [16]

N. Sauer: On the density of families of sets, J. Combin. Th. Ser. A 13 (1972), 145–147.

17. [17]

S. Shelah: A combinatorial problem: Stability and order for models and theories in infinitary languages; Pac. J. Math. 41 (1972), 247–261.

18. [18]

V. N. Vapnik and A. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities, Th. Prob. Applics. 16 (1971), 264–280.

## Author information

Authors

### Corresponding author

Correspondence to Richard P. Anstee.

Research supported in part by NSERC.

Research partially supported by NSERC of first author.

## Rights and permissions

Reprints and Permissions

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