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Graphs and Combinatorics

, Volume 6, Issue 3, pp 287–291 | Cite as

Some new bounds and values for van der Waerden-like numbers

  • Bruce M. Landman
  • Raymond N. Greenwell
Article

Abstract

Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x1,x2,...,x n } that either form an arithmetic progression or for which there exists a polynomialf with integer coefficients and degree exactlyn − 2, andxj+1 =f(x j ). We denote byq(n, k) the least positive integer such that if {1, 2,...,q(n, k)} is partitioned intok classes, then some class must contain a sequence of the type just described. Upper bounds are obtained forq(n, 3), q(n, 4), q(3, k), andq(4, k). A table of several values is also given.

Keywords

Positive Integer Arithmetic Progression Integer Coefficient 
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References

  1. 1.
    Berlekamp, E.R.: A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull.11, 409–414 (1968)Google Scholar
  2. 2.
    Graham, R.L., Rothschild, B.L.: A short proof of van der Waerden's Theorem on arithmetic progressions. Proc. Amer. Math. Soc.42, 385–386 (1974)Google Scholar
  3. 3.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. New York: John Wiley and Sons 1980Google Scholar
  4. 4.
    Greenwell, R.N., Landman, B.M.: On the existence of a reasonable upper bound for the van der Waerden numbers. J. Comb. Theory (A), to appearGoogle Scholar
  5. 5.
    Landman, B.M.: Generalized van der Waerden numbers. Graphs and Combinatorics2, 351–356 (1986)Google Scholar
  6. 6.
    Landman, B.M., Greenwell, R.N.: Values and bounds for Ramsey numbers associated with polynomial iteration. Discrete Math.68, 77–83 (1988)Google Scholar
  7. 7.
    Shelah, S.: Primitive recursive bounds for van der Waerden numbers. J. Amer. Math. Soc.1, 683–697 (1988)Google Scholar
  8. 8.
    Waerden, B.L. van der: Beweis einer Baudetschen Vermutung. Nieuw Arch. Wiskd.15, 212–216 (1927)Google Scholar
  9. 9.
    Waerden, B.L.: van der: How the proof of Baudet's Conjecture was found. Studies in Pure Mathematics (edited by L. Mirsky) pp. 251–260. New York: Academic Press, 1971Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Bruce M. Landman
    • 1
  • Raymond N. Greenwell
    • 2
  1. 1.Department of MathematicsUniversity of North Carolina at GreensboroGreensboroUSA
  2. 2.Department of MathematicsHofstra UniversityHempsteadUSA

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