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

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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 languageEnglish (US)
Pages (from-to)93-101
Number of pages9
JournalBulletin of the Australian Mathematical Society
Volume58
Issue number1
Publication statusPublished - Aug 1 1998
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)

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