Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T10:53:19.917Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Beck, J., ‘A remark concerning arithmetic progressions’, J. Combin. Theory Ser. A 29 (1980), 376–37.CrossRefGoogle Scholar
[2]Bergelson, V., ‘Ergodic Ramsey theory – an update’, in Ergodic theory of Zd-actions (Warwick, 1993–1994), London Math. Soc. Lecture Note Ser. 228 (Cambridge, U.K., 1996), pp. 1–61.Google Scholar
[3]Bergelson, V. and Leibman, A., ‘Polynomial extensions of van der Waerden's and Szemerédi's theorems’, J. Amer. Math. Soc. 9 (1996), 725–753.CrossRefGoogle Scholar
[4]Brown, T.C., ‘On van der Waerden's theorem and a theorem of Paris and Harrington’, J. Combin. Theory Ser. A 30 (1981), 108–111.CrossRefGoogle Scholar
[5]Brown, T.C., Graham, R.L., and Landman, B.M., ‘On the set of differences in van der Waerden's theorem on arithmetic progressions’, Canad. Math. Bull. (to appear).Google Scholar
[6]Furstenberg, H., ‘A polynomial Szemerédi theorem’, in Combinatorics, Paul Erdős is Eighty (Vol. 2) (János Bolyai Math. Soc, Budapest, 1996), pp. 253–269.Google Scholar
[7]Justin, J., ‘Théorème de van der Waerden, Lemme de Brown et demi-groupes répétitifs’, (unpublished preprint), in Journées sur la théorie algébrique des Demi-groupes (Faculté de Sciences de Lyon, 1971).Google Scholar
[8]Landman, B.M., On avoiding arithmetic progressions whose common differences belong to a given small set, J. Comb. Math. Comb. Computing (to appear).Google Scholar
[9]Landman, B.M., ‘On some generalizations of the van der Waerden number w (3)’, (preprint).Google Scholar
[10]van der Waerden, B.L., ‘Beweis einer Baudetschen Vermutung’, Nieuw Arch. Wish. 15 (1927), 212–216.Google Scholar
[11]van der Waerden, B.L., ‘How the proof of Baudet's conjecture was found’, in Studies in Pure Math., (Mirsky, L., Editor) (Academic Press, 1971), pp. 251–260.Google Scholar