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) |
|---|---|
| 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 |
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Monochromatic sequences whose gaps belong to {d, 2d, ..., md}. / Landman, Bruce M.
In: Bulletin of the Australian Mathematical Society, Vol. 58, No. 1, 01.08.1998, p. 93-101.Research output: Contribution to journal › Article
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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.
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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 -