### 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 |

State | Published - Aug 1 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Monochromatic sequences whose gaps belong to {d, 2d, ..., md}.** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Bulletin of the Australian Mathematical Society*, vol. 58, no. 1, pp. 93-101.

}

TY - JOUR

T1 - Monochromatic sequences whose gaps belong to {d, 2d, ..., md}

AU - Landman, Bruce M.

PY - 1998/8/1

Y1 - 1998/8/1

N2 - For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,...,k} such that for some positive integer d, xi+1 - xi ∈ {d, 2d,...,md} for i = 1,..., k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2,...,n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r < m, hm(m + r) ≤ 2c(m + r - 1) + 1, where c = [m/(m - r)]. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

AB - For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,...,k} such that for some positive integer d, xi+1 - xi ∈ {d, 2d,...,md} for i = 1,..., k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2,...,n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r < m, hm(m + r) ≤ 2c(m + r - 1) + 1, where c = [m/(m - r)]. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

UR - http://www.scopus.com/inward/record.url?scp=0032147837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032147837&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032147837

VL - 58

SP - 93

EP - 101

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 1

ER -