### Abstract

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f : Z^{+} → R^{+} define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id : 0 ≤ i ≤ k - 1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

Original language | English (US) |
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Pages (from-to) | 21-35 |

Number of pages | 15 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 60 |

Issue number | 1 |

State | Published - Aug 1 1999 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Brown, T. C., & Landman, B. M. (1999). Monochromatic arithmetic progressions with large differences.

*Bulletin of the Australian Mathematical Society*,*60*(1), 21-35.