### Abstract

We generalize the concept of pattern occurrence in permutations, words or matrices to that in n-dimensional objects, which are basically sets of (n + 1)-tuples. In the case n = 3, we give a possible interpretation of such patterns in terms of bipartite graphs. For zero-box patterns we study vanishing borders related to bipartite Ramsey problems in the case of two dimensions. Also, we study the maximal number of 1’s in binary objects avoiding (in two different senses) a zero-box pattern.

Original language | English |
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Pages (from-to) | 291-299 |

Number of pages | 9 |

Journal | Pure Mathematics and Applications |

Volume | 18 |

Issue number | 3-4 |

Publication status | Published - 2007 |

### Keywords

- pattern occurrence
- permutations
- bipartite graphs

## Cite this

Kitaev, S., & Robbins, J. (2007). On multi-dimensional patterns.

*Pure Mathematics and Applications*,*18*(3-4), 291-299.