Monochromatic arithmetic progressions with large differences

Tom C. Brown, Bruce M. Landman

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Abstract

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f : Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id : 0 ≤ i ≤ k - 1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

Original languageEnglish (US)
Pages (from-to)21-35
Number of pages15
JournalBulletin of the Australian Mathematical Society
Volume60
Issue number1
Publication statusPublished - Aug 1 1999
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)

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