### 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) |
---|---|

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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Australian Mathematical Society*,

*60*(1), 21-35.

**Monochromatic arithmetic progressions with large differences.** / Brown, Tom C.; Landman, Bruce M.

Research output: Contribution to journal › Article

*Bulletin of the Australian Mathematical Society*, vol. 60, no. 1, pp. 21-35.

}

TY - JOUR

T1 - Monochromatic arithmetic progressions with large differences

AU - Brown, Tom C.

AU - Landman, Bruce M.

PY - 1999/8/1

Y1 - 1999/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033177589&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033177589&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033177589

VL - 60

SP - 21

EP - 35

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 1

ER -