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Monochromatic arithmetic progressions with large differences

Published online by Cambridge University Press:  17 April 2009

Tom C. Brown  and
Bruce M. Landman
Tom C. Brown
Affiliation:
Department of Mathematics and StatisticsSimon Fraser UniversityBurnaby, B.C.canada, V5A 1S6
Bruce M. Landman
Affiliation:
Department of Mathematical SciencesUniversity of North Carolina at GreensboroGreensboro, N.C. 27402United States of America
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 21 - 35
Copyright
Copyright © Australian Mathematical Society 1999

References

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