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

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

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.

Original language English (US) 93-101 9 Bulletin of the Australian Mathematical Society 58 1 Published - Aug 1 1998

Integer
Progression
Colouring
Lower bound
Denote
Term

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: Bulletin of the Australian Mathematical Society, Vol. 58, No. 1, 01.08.1998, p. 93-101.

Research output: Contribution to journalArticle

@article{351e35077c274b0ca7f18ece9fd64385,
title = "Monochromatic sequences whose gaps belong to {d, 2d, ..., md}",
abstract = "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.",
author = "Landman, {Bruce M.}",
year = "1998",
month = "8",
day = "1",
language = "English (US)",
volume = "58",
pages = "93--101",
journal = "Bulletin of the Australian Mathematical Society",
issn = "0004-9727",
publisher = "Cambridge University Press",
number = "1",

}

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 -