### Abstract

For m and k positive integers, define a k-term h_{m}-progression to be a sequence of positive integers {x_{1},...,_{k}} such that for some positive integer d, x_{i+1} - x_{i} ∈ {d, 2d,...,md} for i = 1,..., k - 1. Let h_{m}(k) denote the least positive integer n such that for every 2-colouring of {1, 2,...,n} there is a monochromatic h_{m}-progression of length k. Thus, h_{1}(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r < m, h_{m}(m + r) ≤ 2c(m + r - 1) + 1, where c = [m/(m - r)]. We also give a lower bound for h_{m}(k) that has order of magnitude 2k^{2}/m. A precise formula for h_{m}(k) is obtained for all m and k such that k ≤ 3m/2.

Original language | English (US) |
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Pages (from-to) | 93-101 |

Number of pages | 9 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 58 |

Issue number | 1 |

Publication status | Published - Aug 1 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)