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) |
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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 |
DOIs | |
State | Published - Aug 1998 |
Externally published | Yes |
ASJC Scopus subject areas
- General Mathematics