### Abstract

We say that a 0-1 matrix A avoids another 0-1 matrix (pattern) P if no matrix P′ obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17-27; J. Combin. Theory Ser. A 55 (1990) 316-320; Discrete Math. 103 (1992) 233-251) and other papers we investigate n by n 0-1 matrices avoiding a pattern P and the maximal number ex (n, P) of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of ex (n, P) for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered.

Original language | English |
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Pages (from-to) | 266-288 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 111 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 2005 |

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### Keywords

- Extremal problem
- Forbidden submatrix

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**On 0-1 matrices and small excluded submatrices.** / Tardos, G.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 111, no. 2, pp. 266-288. https://doi.org/10.1016/j.jcta.2004.11.015

}

TY - JOUR

T1 - On 0-1 matrices and small excluded submatrices

AU - Tardos, G.

PY - 2005/8

Y1 - 2005/8

N2 - We say that a 0-1 matrix A avoids another 0-1 matrix (pattern) P if no matrix P′ obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17-27; J. Combin. Theory Ser. A 55 (1990) 316-320; Discrete Math. 103 (1992) 233-251) and other papers we investigate n by n 0-1 matrices avoiding a pattern P and the maximal number ex (n, P) of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of ex (n, P) for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered.

AB - We say that a 0-1 matrix A avoids another 0-1 matrix (pattern) P if no matrix P′ obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17-27; J. Combin. Theory Ser. A 55 (1990) 316-320; Discrete Math. 103 (1992) 233-251) and other papers we investigate n by n 0-1 matrices avoiding a pattern P and the maximal number ex (n, P) of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of ex (n, P) for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered.

KW - Extremal problem

KW - Forbidden submatrix

UR - http://www.scopus.com/inward/record.url?scp=22644432259&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22644432259&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2004.11.015

DO - 10.1016/j.jcta.2004.11.015

M3 - Article

VL - 111

SP - 266

EP - 288

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -