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

Published online by Cambridge University Press:  17 April 2009

Bruce M. Landman
Bruce M. Landman
Affiliation:
Department of Mathematical Sciences, University of North Carolina at Greensboro, Greensboro, NC 27412, United States of America e-mail: bmlandma@hamlet.uncg.edu
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Abstract

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For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 93 - 101
Copyright
Copyright © Australian Mathematical Society 1998

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